Pith. sign in

REVIEW 2 major objections 6 minor 124 references

LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks

T0 review · 2 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Physics-informed neural networks fail in hard PDE regimes largely because of their random starting weights, and a learned initialization—built from easy tasks and a layer-unlocking schedule—substantially removes those failures.

desk verdict A credible, modest extension of MAML-style learned initialization for PINNs; the headline 91.5% figure reproduces from the tables, but the easy-task set leaks a borderline-hard case and the appendix overstates variance. read the letter →

arxiv 2607.14233 v1 pith:4EBHIGDH submitted 2026-07-15 cs.LG cs.AI

Nilay Anurag , Shital Adhikari , Taniya Kapoor , Nikhil Muralidhar This is my paper
classification cs.LGcs.AI MSC 68T0765M70
keywords physics-informedneuralnetworkslearnedinitializationmeta-learninggatedlayer-wiseoptimizationPDEextrapolationspectralbiascatastrophicconvergencefailureNavier-Stokes
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

In hard PDE regimes—high-frequency convection, oscillatory Helmholtz, high-Reynolds Navier-Stokes—physics-informed neural networks often converge to trivial or inaccurate solutions. The paper's thesis is that the starting weights, not just the loss formulation, are responsible: a PINN that begins in the wrong basin of its loss landscape fails to escape it. LIGO-PINN therefore learns the initialization itself: it meta-trains on easier configurations of the same PDE family to distill shared physics, and during that meta-training progressively unlocks deeper layers so that shallow features are stabilized first. On extrapolation to unseen hard configurations, the learned initialization reduces mean absolute error by 91.5% on average across six baselines and by 81% over the strongest baseline, and it also improves inverse coefficient recovery. The broader point is that weight initialization is a transportable, one-time investment that can be reused across tasks within a PDE family.

What carries the argument

The carrying mechanism is the gated layer-wise meta-update: at meta-iteration r, layer l receives the meta-gradient only if gl(r)=1{l ≤ ⌊rL/R⌋}, so shallow layers are learned first and deeper layers are progressively unlocked. This gate sits on top of Invariance Encoding, in which k copies of the global weights are adapted for one inner step on k easy tasks and the meta-gradient accumulates over those task losses. The gate is load-bearing: ablations show that removing it (plain meta-learning) or removing both components (random initialization) sharply degrades accuracy, while the full method reduces high-frequency error, indicating mitigated spectral bias.

What would settle it

Re-run the 1D convection extrapolation with the easy set reduced to β∈{5,10,15,20}, excluding the borderline β=25. If the average MAE on β∈{40,50,60,70,80} rises to the level of the random-initialization baseline, then the claimed transfer is largely an artifact of near-hard leakage rather than learned physics. A second check: meta-learn on one PDE family and fine-tune on a different one; the method's usefulness as a general initialization prior would be confirmed only if some transfer survives the family switch.

Watch

Extended reading notes

Core claim

The central claim: random weight initialization is a primary cause of catastrophic PINN failures in hard PDE regimes, and a learned initialization prevents them. The method combines Invariance Encoding—a meta-loop that adapts copies of the weights on easy tasks and distills their shared physics into global weights—with Gated Layer-wise Optimization, a binary gate gl(r)=1{l ≤ ⌊rL/R⌋} that gradually exposes deeper layers to meta-gradients. The learned weights then initialize ordinary PINN training on unseen hard tasks. The paper reports an average 91.5% error reduction across six baselines, 81% over the strongest baseline, smoother loss landscapes, reduced spectral bias, and better inverse coe

Load-bearing premise

The whole extrapolation claim rests on the easy/hard task split being meaningful—the easy set must teach transferable physics and the hard set must be genuinely harder—and the paper itself notes that β=25, included in the easy set, already exhibits hard-task loss dynamics by its own 10K-epoch criterion.

Editorial extensions

If this is right

  • Learned initialization can be reused across multiple hard targets within the same PDE family, so the one-time meta-training cost (about 21% extra GPU-hours) amortizes over several downstream tasks.
  • The method improves extrapolation to harder parameters that were never seen during initialization, not just interpolation within the easy set.
  • It extends to inverse problems: estimating advection and viscosity coefficients improves by 80.2% and 45.9% over the strongest baseline at Reynolds numbers 600–1000.
  • It composes with existing stabilization techniques: initializing an adaptive-sampling baseline with learned weights beats the same baseline with standard random initialization even under extended training.
  • The beneficial effect appears on 3D unstructured geometries as well, not just structured 1D and 2D domains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension the paper leaves unexplored is whether the gating schedule can be made adaptive—for instance, driven by residual norms or spectral-bias measurements rather than a fixed linear schedule—which could make the method self-tuning across PDE families.
  • Because the paper's difficulty criterion is early-loss magnitude at 10K epochs, a testable consequence is that the method's gain should shrink when easy and hard regimes become dynamically similar; removing the borderline task β=25 from the easy set (which the paper notes already shows hard-task dynamics) would test how much of the extrapolation depends on clean separation.
  • The spectral-bias result suggests learned initialization may serve as a cheaper alternative or complement to Fourier-feature embeddings for high-frequency PDEs, an experiment that is directly runnable with the released code.
  • The reported amortization implies a practical workflow: a group that repeatedly solves different parameters of the same PDE family should invest in the initialization once; the paper does not quantify break-even, but the roughly 21% overhead suggests a small number of reuse cycles.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes LIGO-PINN, a learned-initialization framework for PINNs. The method first meta-learns initial network weights on a set of easy PDE configurations (invariance encoding, IE) using a MAML-style inner/outer loop, and then applies a gated layer-wise optimization (GLO) schedule that gradually unlocks deeper layers during meta-updates. The learned weights are then fine-tuned on hard, held-out PDE parameter regimes. The authors evaluate on 1D convection, 2D Helmholtz, 2D incompressible Navier–Stokes (forward and inverse), and a 3D unstructured geometry, comparing against six baselines (vanilla PINN, dynamic loss reweighting, curriculum regularization, R3 sampling, Meta-PDE, HyperLR). They report a 91.5% average MAE reduction over six baselines and 81% over the strongest baseline, and support the claim with loss curves, loss-landscape visualizations, and ablations of IE/GLO.

Significance. If the reported results hold, the paper makes a useful contribution: it demonstrates that weight initialization, rather than only loss reweighting, sampling, or curriculum design, can mitigate catastrophic PINN failures in high-frequency and high-Reynolds regimes. The paper is strong on reproducibility: code is provided, hyperparameters are tabulated per PDE system, and sensitivity analyses are included. The main tables are internally consistent; I independently reproduced the 91.5% and 81% aggregate claims from Tables 1–3. The claimed mechanism—meta-learning an initialization from related easy tasks—is plausible and, if properly validated, would be a practical complement to existing PINN stabilization methods. The limitations are honestly stated (need for related PDE configurations, lack of theoretical analysis). The central risk is not internal arithmetic but the experimental protocol for some baselines and the definition of the easy/hard split.

major comments (2)
  1. [§4 and Appendix B.3–B.4] The evaluation protocol says all baselines are trained for 56K epochs on the target domain and describes Meta-PDE and HyperLR only as meta-learning methods. Meta-PDE (MAML-style) and HyperLR (hypernetwork-generated weights) are not designed to be trained from random initialization on a single target task. If that is what was done, their high errors in Tables 1–4 are not surprising and the comparison is not apples-to-apples. Please specify exactly how Meta-PDE and HyperLR were configured: did they receive any meta-training on ΓEasy, how many meta-iterations, and what adaptation protocol? If they did not, either rerun them under their intended protocols or clearly label them as 'meta-learning baselines without meta-training' and revise the 'six state-of-the-art baselines' claim.
  2. [§4, Appendix B.2, Fig. 6a] Appendix B.2 explicitly states that β=25 exhibits 'hard task dynamics' yet retains it in ΓEasy to match the Curr-Reg setup. This weakens the paper's central easy-to-hard transfer claim: the meta-initialization has seen a near-hard task, so the large gains on β=40–60 in Table 1 may reflect partial task leakage rather than pure extrapolation. The sensitivity analysis in Table 17 does not resolve this, since every task-range variant includes β=25 or harder values. Please add an explicit ablation with ΓEasy = {5,10,15,20} (and, if feasible, ΓEasy = {5,10,15,20,30}) while keeping the same ΓHard test set, and report the resulting Table 1 performance. If performance is unchanged, state so; if not, the extrapolation claim should be correspondingly qualified.
minor comments (6)
  1. [Figure 1 captions] The right-hand panels are labeled 'LeIn-PINN Prediction' but the text and method name refer to 'LIGO-PINN.' Please correct.
  2. [Algorithm 1] The loop line 'for r←0 do // meta-iterations r=1,...,R' is confusing; it should be 'for r=1,...,R do'.
  3. [Figure 6 caption] The caption says 'Distribution of residual losses @10K' but the text describes 'total loss.' Please align terminology.
  4. [§3, Eq. (2) and Appendix A.1] Eq. (2) writes the total loss with λ_r only on the residual term and no explicit weight on the data/boundary loss, while Appendix A.1 Eq. (15) introduces both λ_r and λ_d. Please clarify that the main text implicitly sets λ_d=1, consistent with the reported λ_res:λ_data ratios in Tables 5–7.
  5. [§5.2, 2D Helmholtz paragraph] The text calls PINN-Dynamic the strongest non-ours baseline in 2D Helmholtz, but Table 2 shows Curr-Reg (0.4160) is marginally better than PINN-Dynamic (0.4200) on average. Please correct the phrasing.
  6. [Appendix C.7] Tables 16 and 17 report median MAE, while the main tables report mean MAE. Please state why the median is used in the sensitivity analysis and whether the conclusions also hold for the mean.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: LIGO-PINN's learned initialization is meta-trained on ΓEasy and evaluated on held-out ΓHard; no target quantity is fit, renamed, or forced by self-citation.

full rationale

I walked the derivation chain: the learned initialization ΘR is produced by Algorithm 1 using only tasks sampled from p(ΓEasy) (Section 3, Eq. 3–5), and the hard evaluation tasks β∈{40,...,80}, (a1,a2) high-frequency pairs, and Re∈{600,800,1000} are disjoint from ΓEasy (Section 4, Appendix B.2). The reported MAE tables and the 91.5%/81% headline are descriptive aggregates of held-out comparisons, not quantities used to set the initialization. The gating schedule in Eq. 4 is an explicit hand-chosen linear schedule, not an imported ansatz or uniqueness theorem. I found no self-citations, no fitted parameter renamed as a prediction, and no definitional equivalence between the method's objective and its evaluation metric. The one in-scope caveat is Appendix B.2's admission that 'despite β=25 exhibiting hard task dynamics, it is included in the easy task set to maintain consistency with the experimental setup employed in Curr-Reg.' This is a benchmark-construction confound that may weaken the cleanliness of the easy-to-hard extrapolation claim, and the paper's own Fig. 6a supports that concern. It is not, however, circular: the meta-training loss, the difficulty split, and the evaluation MAE are distinct quantities, and the hard tasks above β=25 remain unseen during invariance encoding. The ablation analyses (RQ3) also compare against random and MAML-style initialization rather than reusing the target result. Overall, the central claim is self-contained and not forced by construction.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central method rests on a small set of hand-chosen hyperparameters (meta-training schedule, loss weights, task split) and on two domain assumptions: transfer from easy PDE tasks to hard ones, and layer-wise gating as a stabilizing mechanism. No new physical entities are introduced.

free parameters (3)
  • Meta-training hyperparameters (R=200 outer steps, J=10 inner steps, k=3 sampled tasks, outer/inner LR per domain)
    Hand-chosen; sensitivity reported only for k and task range, not for learning rates or schedule.
  • Loss weighting λ_res:λ_data = 0.01:1 (1D, Helmholtz); 1:1 (Navier-Stokes)
    Chosen per domain by hand; central to optimizer behavior and not swept.
  • Continuity weight α in NS residual = not reported
    Defined in Eq. 39 but no value is given in main text or appendix; affects the NS residual loss.
assumptions (4)
  • domain assumption PINNs can approximate PDE solutions and automatic differentiation gives correct residuals
    Standard PINN setup from Raissi et al.; invoked throughout Section 3 and Appendix A.
  • domain assumption Meta-learning on Γeasy distills physics that transfers to Γhard
    Core premise of Invariance Encoding, Section 3: a network trained on easy tasks yields useful initialization for hard tasks.
  • ad hoc to paper Early-training loss at 10K epochs is a valid difficulty measure and the chosen easy/hard thresholds are correct
    Appendix B.2/Fig. 6; used to define task splits, including β=25 as easy despite its own hard-task dynamics.
  • ad hoc to paper Gradual layer unmasking (GLO) improves meta-update stability
    Eq. 4-5; no theoretical or empirical comparison against other schedules; only with/without GLO ablation.

how reviews work

0 comments
Cite this review

Pith. "Pith review of LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks." pith.science (2026). https://pith.science/paper/4EBHIGDH

@misc{pith2026260714233,
  author       = {Pith},
  title        = {Pith review of: LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EBHIGDH}},
  note         = {Machine review of arXiv:2607.14233}
}
read the original abstract

Physics-informed neural networks (PINNs) have had a broad research impact in modeling domains governed by partial differential equations (PDE). However, PINNs have been shown to perform poorly, sometimes even converging to trivial solutions, in challenging PDE domains, or when generalizing to unseen but related PDE domains. Previously proposed solutions detail hyperparameter tuning to reduce loss imbalance between data-driven and physics guided losses, curriculum learning based training strategies, or dynamic re-sampling of hard collocation points. These methods face certain pitfalls: hyperparameter tuning is expensive, designing a training curriculum is ambiguous in multi-parameter PDE settings, and dynamic resampling still fails in complex PDE settings. Complementary to this line of thinking, we believe the initial PINN network weights also play a crucial role in the emergence of catastrophic failures during training, yet the effect of PINN weight initialization has been surprisingly under-investigated. To this end, we propose a framework for Learned Initialization via Gated Layerwise Optimization (LIGO-PINN) to overcome PINN convergence failures. Through rigorous evaluation on 1D and 2D PDE domains, including a challenging 2D fluid dynamics setting, we demonstrate that our methodology outperforms state-of-the-art methods designed to alleviate PINN failures, achieving a 91.5% average performance improvement across six baselines and 81% over the strongest baseline. We also verify that LIGO-PINN generalizes to 3D unstructured domains. Finally, we analyze training dynamics across all three PDE domains to explain both LIGO-PINN's improvement and the convergence failure of traditional PINNs. Code: https://github.com/scailab/ligo-pinn Keywords: Machine Learning, Physics-Informed Neural Networks, Deep Learning, PDE Modeling

Figures

Figures reproduced from arXiv: 2607.14233 by the authors.

Figure 1
Figure 1. Qualitative performance comparison of a randomly initialized PINN model and LIGO [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Navier-Stokes cylinder wake at Re = 1000. (a) Simulation setup with vortex shedding (FEM solver). The black box indicates the region used for training and evaluation. (b–c) Velocity components ux, uy at a representative time slice. Also showcased are collocation points at which velocity training data is sampled. (d) Ground-truth pressure. (e) Standard PINN prediction (failure). (f) LIGO-PINN prediction, showing fait… view at source ↗
Figure 3
Figure 3. Loss curves highlighting training dynamics of randomly initialized PINNs and LIGO-PINN, [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Loss Landscape visualization generated by perturbing converged model weights along [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: The figure depicts results of LIGO-PINN ablation analysis on the most sophisticated PDE [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: Distribution of residual losses @ 10K across five random seeds. 1D Convection exhibits [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: Poisson equation on a bunny surface. Overall, we find that the LIGO-PINN achieves a [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Loss Landscape visualization by pertubating model along its top-2 eigenvectors ( [PITH_FULL_IMAGE:figures/full_fig_p024_8.png]
Figure 9
Figure 9. Figure 9: Empirical (MAE) comparison across ablation variants on 1D Convection and 2D Helmholtz. [PITH_FULL_IMAGE:figures/full_fig_p027_9.png]
Figure 10
Figure 10. Figure 10: Spectral distribution of absolute error for LIGO-PINN and ablation variants. Lower [PITH_FULL_IMAGE:figures/full_fig_p027_10.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

124 extracted references · 2 canonical work pages

  1. [1]

    How to initialize your network? robust initialization for weightnorm & resnets

    Devansh Arpit, V \' ctor Campos, and Yoshua Bengio. How to initialize your network? robust initialization for weightnorm & resnets. Advances in Neural Information Processing Systems, 32, 2019

    2019

  2. [3]

    Hypernetwork-based meta-learning for low-rank physics-informed neural networks

    Woojin Cho, Kookjin Lee, Donsub Rim, and Noseong Park. Hypernetwork-based meta-learning for low-rank physics-informed neural networks. Advances in Neural Information Processing Systems, 36: 0 11219--11231, 2023

    2023

  3. [4]

    Scientific machine learning through physics--informed neural networks: Where we are and what’s next

    Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics--informed neural networks: Where we are and what’s next. Journal of Scientific Computing, 92 0 (3): 0 88, 2022 a

    2022

  4. [5]

    Scientific machine learning through physics-informed neural networks: Where we are and what's next, 2022 b

    Salvatore Cuomo, Vincenzo Schiano di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics-informed neural networks: Where we are and what's next, 2022 b . URL https://arxiv.org/abs/2201.05624

    arXiv 2022

  5. [6]

    Mitigating propagation failures in physics-informed neural networks using retain-resample-release (r3) sampling, 2023

    Arka Daw, Jie Bu, Sifan Wang, Paris Perdikaris, and Anuj Karpatne. Mitigating propagation failures in physics-informed neural networks using retain-resample-release (r3) sampling, 2023. URL https://arxiv.org/abs/2207.02338

    arXiv 2023

  6. [7]

    Model- Agnostic Meta - Learning for Fast Adaptation of Deep Networks , July 2017

    Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model- Agnostic Meta - Learning for Fast Adaptation of Deep Networks , July 2017. URL http://arxiv.org/abs/1703.03400. arXiv:1703.03400 [cs]

    arXiv 2017

  7. [8]

    Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the neural tangent kernel

    Stanislav Fort, Gintare Karolina Dziugaite, Mansheej Paul, Sepideh Kharaghani, Daniel M Roy, and Surya Ganguli. Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the neural tangent kernel. Advances in Neural Information Processing Systems, 33: 0 5850--5861, 2020

    2020

  8. [9]

    Flat minima

    Sepp Hochreiter and J \"u rgen Schmidhuber. Flat minima. Neural computation, 9 0 (1): 0 1--42, 1997

    1997

Show all 124 references
  1. [10]

    Dpm: A novel training method for physics-informed neural networks in extrapolation

    Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jhin, and Noseong Park. Dpm: A novel training method for physics-informed neural networks in extrapolation. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 35, pages 8146--8154, 2021

    2021

  2. [11]

    Krishnapriyan, Amir Gholami, Shandian Zhe, Robert M

    Aditi S. Krishnapriyan, Amir Gholami, Shandian Zhe, Robert M. Kirby, and Michael W. Mahoney. Characterizing possible failure modes in physics-informed neural networks, November 2021. URL http://arxiv.org/abs/2109.01050. arXiv:2109.01050 [physics]

    2021 arXiv

  3. [13]

    Physics informed neural networks for fluid flow analysis with repetitive parameter initialization

    Jongmok Lee, Seungmin Shin, Taewan Kim, Bumsoo Park, Ho Choi, Anna Lee, Minseok Choi, and Seungchul Lee. Physics informed neural networks for fluid flow analysis with repetitive parameter initialization. Scientific Reports, 15 0 (1): 0 16740, 2025

    2025

  4. [15]

    Tian Qin, Alex Beatson, Deniz Oktay, Nick McGreivy, and Ryan P. Adams. Meta-pde: Learning to solve pdes quickly without a mesh, 2022. URL https://arxiv.org/abs/2211.01604

    2022 arXiv

  5. [16]

    Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations, 2017 a

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations, 2017 a . URL https://arxiv.org/abs/1711.10566

    2017 arXiv

  6. [18]

    Numerical gaussian processes for time-dependent and nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George Em Karniadakis. Numerical gaussian processes for time-dependent and nonlinear partial differential equations. SIAM Journal on Scientific Computing, 40 0 (1): 0 A172--A198, 2018

    2018

  7. [19]

    Meta-learning with implicit gradients

    Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and Sergey Levine. Meta-learning with implicit gradients. Advances in neural information processing systems, 32, 2019

    2019

  8. [22]

    Dats: Difficulty-aware task sampler for meta-learning physics-informed neural networks

    Maryam Toloubidokhti, Yubo Ye, Ryan Missel, Xiajun Jiang, Nilesh Kumar, Ruby Shrestha, and Linwei Wang. Dats: Difficulty-aware task sampler for meta-learning physics-informed neural networks. In The Twelfth International Conference on Learning Representations, 2023

    2023

  9. [23]

    Zippered polygon meshes from range images

    Greg Turk and Marc Levoy. Zippered polygon meshes from range images. In Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '94, pages 311--318, New York, NY, USA, 1994. ACM. doi:10.1145/192161.192241

    1994

  10. [24]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks

    Sifan Wang, Yujun Teng, and Paris Perdikaris. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing, 43 0 (5): 0 A3055--A3081, 2021 a

    2021

  11. [25]

    On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. On the eigenvector bias of fourier feature networks: From regression to solving multi-scale pdes with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 384: 0 113938, 2021 b

    2021

  12. [27]

    When and why pinns fail to train: A neural tangent kernel perspective

    Sifan Wang, Xinling Yu, and Paris Perdikaris. When and why pinns fail to train: A neural tangent kernel perspective. Journal of Computational Physics, 449: 0 110768, 2022 b

    2022

  13. [28]

    Gradient alignment in physics-informed neural networks: A second-order optimization perspective, 2025

    Sifan Wang, Ananyae Kumar Bhartari, Bowen Li, and Paris Perdikaris. Gradient alignment in physics-informed neural networks: A second-order optimization perspective, 2025. URL https://arxiv.org/abs/2502.00604

    2025

  14. [29]

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks

    Chenxi Wu, Min Zhu, Qinyang Tan, Yadhu Kartha, and Lu Lu. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 403: 0 115671, 2023

    2023

  15. [31]

    Mean field residual networks: On the edge of chaos

    Ge Yang and Samuel Schoenholz. Mean field residual networks: On the edge of chaos. Advances in neural information processing systems, 30, 2017

    2017

  16. [32]

    Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education

    Clancey, William J. Communication, Simulation, and Intelligent Agents: Implications of Personal Intelligent Machines for Medical Education. Proceedings of the Eighth International Joint Conference on Artificial Intelligence (IJCAI-83)

  17. [33]

    Classification Problem Solving

    Clancey, William J. Classification Problem Solving. Proceedings of the Fourth National Conference on Artificial Intelligence

  18. [34]

    , title =

    Robinson, Arthur L. , title =. 1980 , doi =. https://science.sciencemag.org/content/208/4447/1019.full.pdf , journal =

    1980

  19. [35]

    New Ways to Make Microcircuits Smaller---Duplicate Entry

    Robinson, Arthur L. New Ways to Make Microcircuits Smaller---Duplicate Entry. Science

  20. [36]

    Clancey and Glenn Rennels , abstract =

    Diane Warner Hasling and William J. Clancey and Glenn Rennels , abstract =. Strategic explanations for a diagnostic consultation system , journal =. 1984 , issn =. doi:https://doi.org/10.1016/S0020-7373(84)80003-6 , url =

    1984 doi

  21. [37]

    and Rennels, Glenn R

    Hasling, Diane Warner and Clancey, William J. and Rennels, Glenn R. and Test, Thomas. Strategic Explanations in Consultation---Duplicate. The International Journal of Man-Machine Studies

  22. [38]

    Poligon: A System for Parallel Problem Solving

    Rice, James. Poligon: A System for Parallel Problem Solving

  23. [39]

    Transfer of Rule-Based Expertise through a Tutorial Dialogue

    Clancey, William J. Transfer of Rule-Based Expertise through a Tutorial Dialogue

  24. [40]

    The Engineering of Qualitative Models

    Clancey, William J. The Engineering of Qualitative Models

  25. [41]

    2017 , eprint=

    Attention Is All You Need , author=. 2017 , eprint=

    2017

  26. [42]

    Pluto: The 'Other' Red Planet

    NASA. Pluto: The 'Other' Red Planet

  27. [43]

    arXiv preprint arXiv:1711.10561 , year=

    Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations , author=. arXiv preprint arXiv:1711.10561 , year=

    arXiv

  28. [44]

    arXiv preprint arXiv:2404.07662 , year=

    PINNACLE: PINN Adaptive ColLocation and Experimental points selection , author=. arXiv preprint arXiv:2404.07662 , year=

    arXiv

  29. [45]

    Journal of Machine Learning Research , volume=

    Deep hidden physics models: Deep learning of nonlinear partial differential equations , author=. Journal of Machine Learning Research , volume=

  30. [46]

    arXiv preprint arXiv:2403.15989 , year=

    Knowledge-guided Machine Learning: Current Trends and Future Prospects , author=. arXiv preprint arXiv:2403.15989 , year=

    arXiv

  31. [47]

    and Zwart, Jacob A

    Jia, Xiaowei and Willard, Jared and Karpatne, Anuj and Read, Jordan S. and Zwart, Jacob A. and Steinbach, Michael and Kumar, Vipin , title =. 2021 , issue_date =. doi:10.1145/3447814 , journal =

    2021 doi

  32. [48]

    Knowledge Guided Machine Learning , pages=

    Physics-guided neural networks (pgnn): An application in lake temperature modeling , author=. Knowledge Guided Machine Learning , pages=. 2022 , publisher=

    2022

  33. [49]

    and Steinbach, Michael and Banerjee, Arindam and Ganguly, Auroop and Shekhar, Shashi and Samatova, Nagiza and Kumar, Vipin , journal=

    Karpatne, Anuj and Atluri, Gowtham and Faghmous, James H. and Steinbach, Michael and Banerjee, Arindam and Ganguly, Auroop and Shekhar, Shashi and Samatova, Nagiza and Kumar, Vipin , journal=. Theory-Guided Data Science: A New Paradigm for Scientific Discovery from Data , year=

  34. [50]

    2022 , publisher=

    Knowledge guided machine learning: Accelerating discovery using scientific knowledge and data , author=. 2022 , publisher=

    2022

  35. [51]

    , volume =

    Coping with Katrina: Assessing Crisis Management Behaviours in the Big One. , volume =. Journal of Contingencies & Crisis Management , author =. 2010 , keywords =. doi:10.1111/j.1468-5973.2009.00597.x , abstract =

    2010

  36. [52]

    Nobuhito Maru and Takaaki Nomura , year =

  37. [53]

    Publication manual of the American Psychological Association , isbn =

    American Psychological Association , year =. Publication manual of the American Psychological Association , isbn =

  38. [54]

    The Manual of Scientific Style , url =

    Harold Rabinowitz and Suzanne Vogel , publisher =. The Manual of Scientific Style , url =

  39. [55]

    Crisis Management in the Food and Drinks Industry , url =

    Colin Doeg , year =. Crisis Management in the Food and Drinks Industry , url =

  40. [56]

    Incorporating Prior Domain Knowledge into Deep Neural Networks , year=

    Muralidhar, Nikhil and Islam, Mohammad Raihanul and Marwah, Manish and Karpatne, Anuj and Ramakrishnan, Naren , booktitle=. Incorporating Prior Domain Knowledge into Deep Neural Networks , year=

  41. [57]

    and Gholami, Amir and Zhe, Shandian and Kirby, Robert M

    Krishnapriyan, Aditi S. and Gholami, Amir and Zhe, Shandian and Kirby, Robert M. and Mahoney, Michael W. , month = nov, year =. Characterizing possible failure modes in physics-informed neural networks , url =

  42. [58]

    Finn, Chelsea and Abbeel, Pieter and Levine, Sergey , month = jul, year =. Model-. doi:10.48550/arXiv.1703.03400 , abstract =

    doi arXiv

  43. [59]

    arXiv preprint arXiv:1611.01232 , year=

    Deep information propagation , author=. arXiv preprint arXiv:1611.01232 , year=

    arXiv

  44. [60]

    Journal of computational physics , volume=

    The origins of computer weather prediction and climate modeling , author=. Journal of computational physics , volume=. 2008 , publisher=

    2008

  45. [61]

    Journal of Computational Physics , volume=

    Inferring solutions of differential equations using noisy multi-fidelity data , author=. Journal of Computational Physics , volume=. 2017 , publisher=

    2017

  46. [62]

    SIAM Journal on Scientific Computing , volume=

    Numerical Gaussian processes for time-dependent and nonlinear partial differential equations , author=. SIAM Journal on Scientific Computing , volume=. 2018 , publisher=

    2018

  47. [63]

    2023 , eprint=

    Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling , author=. 2023 , eprint=

    2023

  48. [65]

    Meta-learning PINN loss functions , volume=

    Psaros, Apostolos F and Kawaguchi, Kenji and Karniadakis, George Em , year=. Meta-learning PINN loss functions , volume=. doi:10.1016/j.jcp.2022.111121 , journal=

    2022

  49. [66]

    A novel meta-learning initialization method for physics-informed neural networks , volume =

    Liu, Xu and Zhang, Xiaoya and Peng, Wei and Zhou, Weien and Yao, Wen , urldate =. A novel meta-learning initialization method for physics-informed neural networks , volume =. doi:10.1007/s00521-022-07294-2 , abstract =

    doi

  50. [67]

    , urldate =

    Penwarden, Michael and Zhe, Shandian and Narayan, Akil and Kirby, Robert M. , urldate =. Multifidelity Modeling for Physics-Informed Neural Networks (. doi:10.1016/j.jcp.2021.110844 , abstract =

    2021

  51. [68]

    On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type

    Shin, Yeonjong and Darbon, Jerome and Karniadakis, George Em , urldate =. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type. doi:10.4208/cicp.OA-2020-0193 , abstract =

    2020 doi

  52. [69]

    Gradient-enhanced physics-informed neural networks for forward and inverse

    Yu, Jeremy and Lu, Lu and Meng, Xuhui and Karniadakis, George Em , urldate =. Gradient-enhanced physics-informed neural networks for forward and inverse. doi:10.1016/j.cma.2022.114823 , abstract =

    2022

  53. [70]

    2017 , eprint=

    Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations , author=. 2017 , eprint=

    2017

  54. [71]

    Advances in Neural Information Processing Systems , volume=

    Hypernetwork-based meta-learning for low-rank physics-informed neural networks , author=. Advances in Neural Information Processing Systems , volume=

  55. [72]

    Belbute-Peres, Filipe de Avila and Sha, Fei and Chen, Yi-fan , langid =

  56. [73]

    Science Guided Machine Learning: Incorporating Scientific Domain Knowledge for Learning Under Data Paucity & Noisy Contexts , author =

  57. [74]

    and Toscano, Juan Diego and Stergiopulos, Nikolaos and Karniadakis, George Em , urldate =

    Anagnostopoulos, Sokratis J. and Toscano, Juan Diego and Stergiopulos, Nikolaos and Karniadakis, George Em , urldate =. Residual-based attention in physics-informed neural networks , volume =. doi:10.1016/j.cma.2024.116805 , abstract =

    2024

  58. [75]

    and Tegmark, Max , urldate =

    Liu, Ziming and Wang, Yixuan and Vaidya, Sachin and Ruehle, Fabian and Halverson, James and Soljačić, Marin and Hou, Thomas Y. and Tegmark, Max , urldate =

  59. [76]

    2022 , eprint=

    Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next , author=. 2022 , eprint=

    2022

  60. [77]

    Du, Yunshu and Czarnecki, Wojciech M and Jayakumar, Siddhant M and Farajtabar, Mehrdad and Pascanu, Razvan and Lakshminarayanan, Balaji , langid =

  61. [78]

    Proceedings of the AAAI Conference on Artificial Intelligence , volume=

    DPM: A novel training method for physics-informed neural networks in extrapolation , author=. Proceedings of the AAAI Conference on Artificial Intelligence , volume=

  62. [79]

    2024 , eprint=

    Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks , author=. 2024 , eprint=

    2024

  63. [80]

    Advances in neural information processing systems , volume=

    Gradient surgery for multi-task learning , author=. Advances in neural information processing systems , volume=

  64. [81]

    , urldate =

    Penwarden, Michael and Zhe, Shandian and Narayan, Akil and Kirby, Robert M. , urldate =. A metalearning approach for Physics-Informed Neural Networks (. doi:10.1016/j.jcp.2023.111912 , shorttitle =

    2023

  65. [82]

    Advances in neural information processing systems , volume=

    Mean field residual networks: On the edge of chaos , author=. Advances in neural information processing systems , volume=

  66. [83]

    2017 , eprint=

    Deep Information Propagation , author=. 2017 , eprint=

    2017

  67. [84]

    Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given

    Iwata, Tomoharu and Tanaka, Yusuke and Ueda, Naonori , urldate =. Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given

  68. [85]

    2016 , eprint=

    Exponential expressivity in deep neural networks through transient chaos , author=. 2016 , eprint=

    2016

  69. [86]

    Mean Field Theory and its application to deep learning , abstract =

    Pandey, Mudit and Xu, Mimee , langid =. Mean Field Theory and its application to deep learning , abstract =

  70. [87]

    Book III, Lemma V, Case , volume=

    Principia mathematica , author=. Book III, Lemma V, Case , volume=

  71. [88]

    Reviews of Modern Physics , volume=

    Hamilton's principle and the conservation theorems of mathematical physics , author=. Reviews of Modern Physics , volume=. 1951 , publisher=

    1951

  72. [89]

    SIAM Journal on Scientific Computing , volume=

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks , author=. SIAM Journal on Scientific Computing , volume=. 2021 , publisher=

    2021

  73. [90]

    Computer Methods in Applied Mechanics and Engineering , volume=

    A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2023 , publisher=

    2023

  74. [91]

    arXiv preprint arXiv:2203.07404 , year=

    Respecting causality is all you need for training physics-informed neural networks , author=. arXiv preprint arXiv:2203.07404 , year=

    arXiv

  75. [92]

    Optics express , volume=

    Physics-informed neural networks for inverse problems in nano-optics and metamaterials , author=. Optics express , volume=. 2020 , publisher=

    2020

  76. [93]

    Journal of Computational Physics , volume=

    NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations , author=. Journal of Computational Physics , volume=. 2021 , publisher=

    2021

  77. [94]

    Neurocomputing , volume=

    Distributed learning machines for solving forward and inverse problems in partial differential equations , author=. Neurocomputing , volume=. 2021 , publisher=

    2021

  78. [95]

    Computer Methods in Applied Mechanics and Engineering , volume=

    On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks , author=. Computer Methods in Applied Mechanics and Engineering , volume=. 2021 , publisher=

    2021

  79. [96]

    Journal of Computational Physics , volume=

    When and why PINNs fail to train: A neural tangent kernel perspective , author=. Journal of Computational Physics , volume=. 2022 , publisher=

    2022

  80. [97]

    Journal of Scientific Computing , volume=

    Scientific machine learning through physics--informed neural networks: Where we are and what’s next , author=. Journal of Scientific Computing , volume=. 2022 , publisher=

    2022

  81. [98]

    Ihlenburg and I

    F. Ihlenburg and I. Babuška , keywords =. Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM , journal =. 1995 , issn =. doi:https://doi.org/10.1016/0898-1221(95)00144-N , url =

    1995 doi

  82. [99]

    2006 , publisher=

    Generalized inverses: theory and applications , author=. 2006 , publisher=

    2006

  83. [100]

    Advances in neural information processing systems , volume=

    Meta-learning with implicit gradients , author=. Advances in neural information processing systems , volume=

  84. [101]

    arXiv preprint arXiv:1710.11622 , year=

    Meta-learning and universality: Deep representations and gradient descent can approximate any learning algorithm , author=. arXiv preprint arXiv:1710.11622 , year=

    arXiv

  85. [102]

    arXiv preprint arXiv:1803.02999 , year=

    On first-order meta-learning algorithms , author=. arXiv preprint arXiv:1803.02999 , year=

    arXiv

  86. [103]

    Proceedings of the thirteenth international conference on artificial intelligence and statistics , pages=

    Understanding the difficulty of training deep feedforward neural networks , author=. Proceedings of the thirteenth international conference on artificial intelligence and statistics , pages=. 2010 , organization=

    2010

  87. [104]

    Advances in neural information processing systems , volume=

    On the number of linear regions of deep neural networks , author=. Advances in neural information processing systems , volume=

  88. [105]

    Conference on learning theory , pages=

    The power of depth for feedforward neural networks , author=. Conference on learning theory , pages=. 2016 , organization=

    2016

  89. [106]

    arXiv preprint arXiv:1509.08101 , year=

    Representation benefits of deep feedforward networks , author=. arXiv preprint arXiv:1509.08101 , year=

    arXiv

  90. [107]

    Theoretical and Applied Mechanics Letters , volume=

    Physics-informed deep learning for incompressible laminar flows , author=. Theoretical and Applied Mechanics Letters , volume=. 2020 , publisher=

    2020

  91. [108]

    Computational Mechanics , volume=

    Machine learning for metal additive manufacturing: predicting temperature and melt pool fluid dynamics using physics-informed neural networks , author=. Computational Mechanics , volume=. 2021 , publisher=

    2021

  92. [109]

    Frontiers in Physics , volume=

    Physics-informed neural networks for cardiac activation mapping , author=. Frontiers in Physics , volume=. 2020 , publisher=

    2020

  93. [110]

    Medical Image Analysis , volume=

    Physics-informed neural networks for myocardial perfusion MRI quantification , author=. Medical Image Analysis , volume=. 2022 , publisher=

    2022

  94. [111]

    Journal of chemical information and modeling , volume=

    PiNN: A python library for building atomic neural networks of molecules and materials , author=. Journal of chemical information and modeling , volume=. 2020 , publisher=

    2020

  95. [112]

    Computational Materials Science , volume=

    Development of a physically-informed neural network interatomic potential for tantalum , author=. Computational Materials Science , volume=. 2022 , publisher=

    2022

  96. [113]

    arXiv preprint arXiv:2109.11452 , year=

    Revisit geophysical imaging in a new view of physics-informed generative adversarial learning , author=. arXiv preprint arXiv:2109.11452 , year=

    arXiv

  97. [114]

    SEG International Exposition and Annual Meeting , pages=

    Wave propagation with physics informed neural networks , author=. SEG International Exposition and Annual Meeting , pages=. 2020 , organization=

    2020

  98. [115]

    arXiv preprint arXiv:2105.02939 , year=

    Pce-pinns: Physics-informed neural networks for uncertainty propagation in ocean modeling , author=. arXiv preprint arXiv:2105.02939 , year=

    arXiv

  99. [116]

    Neural Networks , volume=

    Tackling the curse of dimensionality with physics-informed neural networks , author=. Neural Networks , volume=. 2024 , publisher=

    2024

  100. [117]

    arXiv preprint arXiv:2010.08895 , year=

    Fourier neural operator for parametric partial differential equations , author=. arXiv preprint arXiv:2010.08895 , year=

    2010 arXiv

  101. [118]

    2020 , eprint=

    Gradient Surgery for Multi-Task Learning , author=. 2020 , eprint=

    2020

  102. [119]

    2021 , eprint=

    Physics-Informed Neural Network Method for Solving One-Dimensional Advection Equation Using PyTorch , author=. 2021 , eprint=

    2021

  103. [120]

    Visualizing high-dimensional loss landscapes with Hessian directions , volume=

    Böttcher, Lucas and Wheeler, Gregory , year=. Visualizing high-dimensional loss landscapes with Hessian directions , volume=. Journal of Statistical Mechanics: Theory and Experiment , publisher=. doi:10.1088/1742-5468/ad13fc , number=

    doi

  104. [121]

    2023 , eprint=

    Multi-Grade Deep Learning , author=. 2023 , eprint=

    2023

  105. [122]

    Advances in Neural Information Processing Systems , volume=

    Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the neural tangent kernel , author=. Advances in Neural Information Processing Systems , volume=

  106. [123]

    Advances in Neural Information Processing Systems , volume=

    How to initialize your network? robust initialization for weightnorm & resnets , author=. Advances in Neural Information Processing Systems , volume=

  107. [124]

    2022 , eprint=

    Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh , author=. 2022 , eprint=

    2022

  108. [125]

    2025 , eprint=

    Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective , author=. 2025 , eprint=

    2025

  109. [126]

    arXiv preprint arXiv:1709.02956 , year=

    Deep residual networks and weight initialization , author=. arXiv preprint arXiv:1709.02956 , year=

    arXiv

  110. [127]

    arXiv preprint arXiv:1706.10239 , year=

    Towards understanding generalization of deep learning: Perspective of loss landscapes , author=. arXiv preprint arXiv:1706.10239 , year=

    arXiv

  111. [128]

    Neural computation , volume=

    Flat minima , author=. Neural computation , volume=. 1997 , publisher=

    1997

  112. [129]

    Advances in neural information processing systems , volume=

    Simplifying neural nets by discovering flat minima , author=. Advances in neural information processing systems , volume=

  113. [130]

    Scientific Reports , volume=

    Physics informed neural networks for fluid flow analysis with repetitive parameter initialization , author=. Scientific Reports , volume=. 2025 , publisher=

    2025

  114. [131]

    The Twelfth International Conference on Learning Representations , year=

    Dats: Difficulty-aware task sampler for meta-learning physics-informed neural networks , author=. The Twelfth International Conference on Learning Representations , year=

  115. [132]

    arXiv preprint arXiv:2503.07070 , year=

    PIED: Physics-Informed Experimental Design for Inverse Problems , author=. arXiv preprint arXiv:2503.07070 , year=

    arXiv

  116. [133]

    Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques , series =

    Turk, Greg and Levoy, Marc , title =. Proceedings of the 21st Annual Conference on Computer Graphics and Interactive Techniques , series =. 1994 , pages =

    1994

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.