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REVIEW 3 major objections 6 minor 78 references

Fourier-enhanced Neural Networks For Systems Biology Applications

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Fourier-enhanced network beats PINN on all six systems biology models

desk verdict A plausible engineering combination for oscillatory ODE/PDEs, but the paper's headline efficiency claim is contradicted by its own Table 4.3 and the accuracy comparison is missing key tuning details. read the letter →

arxiv 2502.07129 v1 pith:YCCBZPKL submitted 2025-02-10 cs.LG q-bio.QM

Enze Xu , Minghan Chen This is my paper · ORCID
classification cs.LGq-bio.QM
keywords systemsbiologyphysics-informedneuralnetworksFouriernetworkadaptiveactivationfunctionvarianceconstraintoscillatorydynamicsrepressilatorTuringpatterns
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

SB-FNN is proposed as an alternative to physics-informed neural networks for solving the differential-equation models used in systems biology. It combines a Fourier neural network architecture with a per-layer adaptive activation function and a variance-based oscillation penalty. Across six cellular and population models, the paper reports that SB-FNN is both more accurate, as measured by normalized mean square error on held-out time points, and more efficient than PINN, and that PINN fails outright on two of the six. If the comparison holds, SB-FNN gives systems biologists a meshless surrogate that captures oscillatory dynamics without labeled ground-truth solutions.

What carries the argument

The central object is the SB-FNN architecture: an input network maps time and space coordinates to a latent space, a stack of Fourier layers applies a fast Fourier transform, keeps $M=12$ modes, multiplies by a learned weight tensor, transforms back, adds a convolutional path, then applies an activation, and an output network maps the last latent layer to the predicted dynamics. The two added mechanisms are the adaptive activation function, a softmax mixture of Tanh, ReLU, Softplus, ELU, GELU and Sin with a trainable weight vector per Fourier layer, and the variance constraint $\Phi(x)=\frac{1}{2}(-\tanh((x-\alpha)\tau)+1)$, which adds a penalty near 1 when a normalized predicted trajectory has near-zero variance and near 0 once variance exceeds threshold $\alpha$. The loss combines initial-condition, residual, boundary, and penalty terms, $\mathcal{L}=\lambda_o\|\hat{y}_0-y_0\|^2+\lambda_f\|\hat{y}_t-G[\hat{y}]\|+\lambda_b\|\hat{y}-g\|^2+\lambda_p P(\hat{y})$, and is minimized by Adam with a decaying learning rate.

What would settle it

Reproduce the two repressilator experiments with the exact penalty parameters used in the paper, or with a documented training-only selection rule, and compare held-out N-MSE to vanilla PINN under identical epoch counts; if the reported advantage disappears or reverses, the claimed gain was test-set fitting. Also run the variance-constraint ablation across several reasonable threshold values to see whether the improvement is robust to that choice.

Watch

Extended reading notes

Core claim

The paper claims that a physics-informed surrogate built from Fourier layers, SB-FNN, solves the initial-value ordinary and partial differential equation systems that arise in systems biology more accurately and more cheaply than the standard PINN. On all six benchmark models, two repressilator gene-circuit variants, SIR and age-structured SIR, and 1D and 2D Turing reaction-diffusion systems, SB-FNN reports lower test N-MSE than PINN, including two cases where PINN effectively fails, and it needs less wall-clock time per epoch as model complexity grows. The two bespoke components, a per-Fourier-layer adaptive activation function that mixes six standard nonlinearities with softmax weights, and a variance penalty that pushes trajectories away from flat, non-oscillatory solutions, are each shown by ablation to improve accuracy on oscillatory systems.

Load-bearing premise

The load-bearing premise is that the two tunable parameters of the oscillation penalty were selected without using the test set; the paper reports neither their values nor the selection procedure.

Editorial extensions

If this is right

  • On all six benchmark models, SB-FNN reports lower test N-MSE than PINN; on Rep6 and 1D Turing, PINN fails to produce usable predictions while SB-FNN does.
  • Because SB-FNN is trained from the governing equations plus initial and boundary conditions rather than from simulated ground truth, it can act as a surrogate solver for biological models without labeled solution data.
  • Per-epoch training time for SB-FNN stays roughly flat as spatial complexity grows, while PINN time rises sharply; in the 2D Turing case PINN takes more than three times as long per epoch.
  • The adaptive activation function alone achieves the best mean accuracy ranking across the six models, and adding the variance constraint further improves accuracy on the two oscillatory repressilator models.

Reading between the lines

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

  • Because the variance penalty is a continuous, differentiable addition to the loss, it could be grafted onto any PINN-style surrogate, not only Fourier architectures; the paper only demonstrates it inside SB-FNN.
  • The paper lists inverse parameter estimation as future work, so a direct extension is to use SB-FNN to fit rate constants and unknown dynamics from observed trajectories, with oscillatory systems as the natural test bed.
  • A comparison against enhanced PINN variants, such as gradient-enhanced or domain-decomposed PINNs, would isolate how much of the reported advantage comes from the Fourier architecture itself rather than from the choice of a vanilla PINN baseline.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The manuscript proposes SB-FNN, a Fourier-neural-network-based solver for systems biology ODE/PDE initial-value problems. The method combines an embedded Fourier layer architecture, a softmax-weighted adaptive activation function, and a variance penalty intended to encourage oscillatory solutions. The paper evaluates SB-FNN against a standard PINN on six models—two repressilator variants, SIR, age-structured SIR, and 1D/2D Turing systems—and reports that SB-FNN is more accurate and more efficient than PINN in all cases. The central claim is therefore that SB-FNN is a reliable and faster physics-informed surrogate for oscillatory and multiscale biological models.

Significance. The target problem is practically important: systems biology models are often oscillatory, multiscale, and expensive to solve, and physics-informed surrogates that improve on PINN in this domain would be useful. The paper has several positive features: it evaluates six models spanning ODE and PDE dynamics, it includes an ablation isolating the variance constraint, and it openly states a limitation concerning stiff dynamics in the conclusion. The variance-constraint idea is a plausible mechanism for helping FNN-type solvers preserve oscillations. However, the headline contribution is currently not established. The efficiency half of the claim is contradicted or unsupported by the paper's own Table 4.3, the accuracy metric excludes components that go to zero in a way that is not justified, and the key variance-penalty hyperparameters are never reported. These are fixable within the scope of the manuscript, but they are load-bearing rather than cosmetic.

major comments (3)
  1. [§4.3.2, Table 4.3] The efficiency claim is not supported by the reported data. As printed, Table 4.3 shows that SB-FNN has a larger per-epoch time than PINN on Rep3 (9.07 s vs 7.10 s), SIR (9.05 s vs 6.95 s), and 2D Turing (4.40 s vs 1.54 s), and the other rows do not show a consistent SB-FNN advantage. Since both methods are run to the same maximum number of epochs, a larger per-epoch time implies a slower total run unless SB-FNN converges in substantially fewer epochs, and no epoch-to-accuracy or time-to-accuracy comparison is reported. Moreover, the text's claim that PINN takes more than three times as much time as SB-FNN on 2D Turing is contradicted by the table, which shows the opposite ordering. The authors should replace this table with total training time or a time-to-target-accuracy comparison, or remove the efficiency claim.
  2. [§4.1, Eq. (4.2)] The decision to exclude variables whose dynamics reach zero from the N-MSE computation is not justified and is consequential for the SIR and A-SIR results. In those models the infectious compartment decays to zero, so excluding zero-reaching dynamics removes precisely the part of the solution that is hardest to fit and that carries important qualitative information. As written, the reported accuracy for SIR and A-SIR does not measure the full solution. The authors should report N-MSE per variable or use a stabilized denominator, and should show whether the qualitative conclusion changes when the omitted dynamics are included.
  3. [§3.3, Eq. (3.15) and §4.1, Table 4.1] The variance penalty depends on threshold α and slope τ, and Eqs. (3.5) and (3.14) depend on loss weights λo, λf, λb, and λp. The paper states that α and τ are 'tailored to different systems biology models' but never reports their values, the search grid, or the selection criterion, and the λ weights are likewise absent from Table 4.1. Because the only gains on the oscillatory repressilator models come from this penalty (Table 4.6), the accuracy comparison is not reproducible and may reflect selection against the test N-MSE. The authors should report all hyperparameters per model and describe a selection procedure that does not use the test metric.
minor comments (6)
  1. [Table 4.3] The numeric formatting of Table 4.3 is ambiguous: entries such as '1.0262 ± 0.0395(e−1)' for Rep6 are difficult to parse because the exponent appears only in the error term and the main value may be missing a digit. Use a single, explicit decimal format for both the mean and the standard deviation.
  2. [§4.3.1, Figure 4.8] The caption and text refer to 'training loss (N-MSE)', but Eq. (4.2) defines N-MSE against ground truth, which is not used in training. Clarify whether the plotted quantity is N-MSE evaluated on the training points at each epoch, or a physics residual loss that has been rescaled.
  3. [Eq. (3.5)] The residual term in Eq. (3.5) is written as ∥ ŷ_t − G[ŷ]∥ without the square that appears in the loss diagram in Figure 3.1(b); align the equation with the figure and with the other loss terms.
  4. [§3.2, Eq. (3.11)] The notation Sin∗(x) is introduced but not used afterward, and the scaling factor β in Eq. (3.11) is separate from the variance-penalty threshold α introduced in Eq. (3.15); using distinct symbols is fine, but the unused notation should be removed for clarity.
  5. [Abstract and Conclusion] The closing sentence of the abstract and the conclusion that SB-FNN 'is expected to replace PINN as the most advanced method in systems biology' is an unsupported extrapolation from six benchmark models and should be replaced with a statement about the scope of the evidence.
  6. [§4.1] No comparison is made with a conventional ODE/PDE solver or with other operator-learning baselines; adding a classical solver reference would help calibrate both the accuracy and the efficiency results against the standard workflow that the paper claims to accelerate.

Circularity Check

1 steps flagged · score 3.0 of 10

The core benchmark is empirical rather than derivational, but the variance-constraint accuracy gain on oscillatory models is a fitted input presented as a predictive improvement; the efficiency claim is contradicted by the paper's own Table 4.3.

  1. fitted input called prediction [Sec. 3.3 (Eqs. 3.14-3.15) and Sec. 4.5 (Table 4.6)]
    "Lossp(ŷ) = λp · P(ŷ) = λp · Σ_{i=1}^D Φ(Var(Norm(ŷ[i,:]))) ... Φ(x) = 1/2 · (−tanh((x−α)·τ)+1) ... The hyper-parameters α, τ need to be tailored to different systems biology models."

    The variance penalty is defined so that Φ approaches 1 for low-variance predictions and drops toward 0 once variance exceeds the model-specific threshold α. When this penalty is added to the training loss in Eq. 3.5, any non-oscillatory solution is penalized by construction. The two models where the constraint is evaluated, Rep3 and Rep6, have oscillatory ground truth with large variance, so the N-MSE improvements reported in Sec. 4.5 are at least partly an artifact of inserting 'oscillation' into the objective rather than a discovered predictive property. Because α and τ are tailored per model and neither their values nor the selection procedure are reported, the comparison does not rule out test-set fitting.

full rationale

There is no derivation in the paper that reduces a predicted quantity to a fitted quantity by identity, and the main comparison to PINN is an empirical benchmark on six models, so the central accuracy claim retains independent content. The one partial circularity is the variance constraint: its functional form and its per-model tailoring encode the oscillatory outcome that is then reported as improved prediction of oscillatory dynamics. That is a fitted input called a prediction, though it affects only the Rep3 and Rep6 component of the results. Separately, the efficiency half of the headline is not circular but is unsupported and internally inconsistent: Sec. 4.3.2 states 'In the 2D Turing model, PINN takes more than three times as much time as SB-FNN,' yet Table 4.3 lists 1.5404 s/epoch for PINN versus 4.4034 s/epoch for SB-FNN, and no total training time or time-to-accuracy comparison is reported. The conclusion also acknowledges a limitation on stiff dynamics, which further qualifies the 'replace PINN' claim. Overall, the circularity score is modest because the central claim does not reduce to a self-citation chain or to an equation identity; it is weakened by one tuned loss component and by missing efficiency evidence.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central empirical claim rests on two unreported hyperparameter sets (loss weights and variance penalty parameters), standard benchmark models, and an unspecified numerical reference. No new physical entities are introduced.

free parameters (2)
  • Loss weights λ_o, λ_f, λ_b, λ_p = not reported
    Balance initial, residual, boundary, and variance-penalty terms in Eq. 3.5; without values the optimized objective is underspecified.
  • Variance penalty threshold α and slope τ = not reported
    Eq. 3.15 sets when low variance is penalized; §3.3 says they must be tailored per model, but no values are given.
assumptions (5)
  • domain assumption The six benchmark models in Appendix A (Repressilator, SIR, A-SIR, Schnakenberg Turing) are valid descriptions of the target dynamics.
    The method's accuracy is judged against numerical solutions of these equations; misspecified models would make the benchmark meaningless.
  • domain assumption The numerical ground truth used for N-MSE is accurate; no solver, tolerance, or discretization is specified.
    §4.1 and §4.2 compare predictions with 'calculated' ground truth but never name the solver.
  • domain assumption The physics-informed loss in Eq. 3.5 is a valid objective for solving the initial and boundary value problem.
    Standard PINN assumption: minimizing residual plus initial and boundary terms yields the solution.
  • domain assumption Fourier layers with 12 modes can represent the target dynamics well enough.
    No convergence theorem is given; the empirical N-MSE is the only support.
  • ad hoc to paper Low-variance normalized trajectories should be penalized for oscillatory systems.
    The variance penalty is a hand-designed inductive bias, not derived from the governing equations.

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Cite this review

Pith. "Pith review of Fourier-enhanced Neural Networks For Systems Biology Applications." pith.science (2026). https://pith.science/paper/YCCBZPKL

@misc{pith2026250207129,
  author       = {Pith},
  title        = {Pith review of: Fourier-enhanced Neural Networks For Systems Biology Applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCCBZPKL}},
  note         = {Machine review of arXiv:2502.07129}
}
read the original abstract

In the field of systems biology, differential equations are commonly used to model biological systems, but solving them for large-scale and complex systems can be computationally expensive. Recently, the integration of machine learning and mathematical modeling has offered new opportunities for scientific discoveries in biology and health. The emerging physics-informed neural network (PINN) has been proposed as a solution to this problem. However, PINN can be computationally expensive and unreliable for complex biological systems. To address these issues, we propose the Fourier-enhanced Neural Networks for systems biology (SB-FNN). SB-FNN uses an embedded Fourier neural network with an adaptive activation function and a cyclic penalty function to optimize the prediction of biological dynamics, particularly for biological systems that exhibit oscillatory patterns. Experimental results demonstrate that SB-FNN achieves better performance and is more efficient than PINN for handling complex biological models. Experimental results on cellular and population models demonstrate that SB-FNN outperforms PINN in both accuracy and efficiency, making it a promising alternative approach for handling complex biological models. The proposed method achieved better performance on six biological models and is expected to replace PINN as the most advanced method in systems biology.

Figures

Figures reproduced from arXiv: 2502.07129 by the authors.

Figure 3.1
Figure 3.1. Fourier-enhanced Neural Network for dynamic systems biology model [PITH_FULL_IMAGE:figures/full_fig_p020_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Adaptive activation function. The adaptive weights, denoted as [PITH_FULL_IMAGE:figures/full_fig_p026_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Plot of the variance penalty function Φ ( [PITH_FULL_IMAGE:figures/full_fig_p028_3_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4.1
Figure 4.1. Figure 4.1: Prediction of the Repressilator: protein only model. [PITH_FULL_IMAGE:figures/full_fig_p033_4_1.png]
Figure 4.2
Figure 4.2. Figure 4.2: Prediction of the Repressilator: mRNA and protein model. [PITH_FULL_IMAGE:figures/full_fig_p034_4_2.png]
Figure 4.3
Figure 4.3. Figure 4.3: Prediction of the SIR model. S, I, R represents the susceptible, infectious, and recovered population, respectively. The model parameters are β = 0.01, γ = 0.05 and N = 100. the complete definition and equations for this model. The prediction results for this model u…
Figure 4.4
Figure 4.4. Figure 4.4: Prediction of the Age-structured SIR model. [PITH_FULL_IMAGE:figures/full_fig_p036_4_4.png]
Figure 4.5
Figure 4.5. Figure 4.5: Prediction of the 1D Turing model. U and V are the concentrations of two diffusible substances. This model is generated on a plane space of size 100 × 1 (X represents the first dimension) and a time domain of [0, 10]. It is parameterized using c1 = 0.1, c2 = 0.9, c−1…
Figure 4.6
Figure 4.6. Figure 4.6: Prediction of the 2D Turing model on the last time step. [PITH_FULL_IMAGE:figures/full_fig_p038_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Performance comparison between PINN and SB-FNN on different systems [PITH_FULL_IMAGE:figures/full_fig_p039_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Comparison of training loss (N-MSE) between PINN and SB-FNN over [PITH_FULL_IMAGE:figures/full_fig_p040_4_8.png]
Figure 4.9
Figure 4.9. Figure 4.9: Mean score of different activation functions in descending order. For each [PITH_FULL_IMAGE:figures/full_fig_p044_4_9.png]
Figure 4.10
Figure 4.10. Figure 4.10: Accuracy comparison of methods with and without variance constraint. [PITH_FULL_IMAGE:figures/full_fig_p045_4_10.png]

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Reference graph

Works this paper leans on

78 extracted references · 66 canonical work pages

  1. [1]

    Computational modeling, formal analysis, and tools for systems biology

    Ezio Bartocci and Pietro Li´ o. Computational modeling, formal analysis, and tools for systems biology. PLoS computational biology, 12(1):e1004591, 2016

    work page 2016

  2. [2]

    Stably accelerating stiff quantitative systems pharmacol- ogy models: Continuous-time echo state networks as implicit machine learning

    Ranjan Anantharaman, Anas Abdelrehim, Anand Jain, Avik Pal, Danny Sharp, Chris Rackauckas, et al. Stably accelerating stiff quantitative systems pharmacol- ogy models: Continuous-time echo state networks as implicit machine learning. bioRxiv, 2021

    work page 2021

  3. [3]

    Multipole graph neu- ral operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Andrew Stuart, Kaushik Bhattacharya, and Anima Anandkumar. Multipole graph neu- ral operator for parametric partial differential equations. Advances in Neural Information Processing Systems, 33:6755–6766, 2020

    2020

  4. [4]

    Systems biology: a brief overview

    Hiroaki Kitano. Systems biology: a brief overview. science, 295(5560):1662– 1664, 2002

    work page 2002

  5. [5]

    An introduction to systems biology: design principles of biological circuits

    Uri Alon. An introduction to systems biology: design principles of biological circuits. CRC press, 2019

    work page 2019

  6. [6]

    Integrating high-throughput and computational data elu- cidates bacterial networks

    Markus W Covert, Eric M Knight, Jennifer L Reed, Markus J Herrgard, and Bernhard O Palsson. Integrating high-throughput and computational data elu- cidates bacterial networks. Nature, 429(6987):92–96, 2004. 40

    work page 2004

  7. [7]

    Cell-signalling dynamics in time and space

    Boris N Kholodenko. Cell-signalling dynamics in time and space. Nature reviews Molecular cell biology , 7(3):165–176, 2006

    work page 2006

  8. [8]

    Stochastic vs

    Ranjan Srivastava, L You, J Summers, and J Yin. Stochastic vs. deterministic modeling of intracellular viral kinetics. Journal of theoretical biology, 218(3):309– 321, 2002

    work page 2002

Show all 78 references
  1. [9]

    Universal differential equations for scientific machine learning

    Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, and Alan Edelman. Universal differential equations for scientific machine learning. arXiv preprint arXiv:2001.04385, 2020

    2001 arXiv

  2. [10]

    Explain- able machine learning for scientific insights and discoveries

    Ribana Roscher, Bastian Bohn, Marco F Duarte, and Jochen Garcke. Explain- able machine learning for scientific insights and discoveries. Ieee Access, 8:42200– 42216, 2020

    2020

  3. [11]

    Machine learning for quantum mechanics in a nutshell

    Matthias Rupp. Machine learning for quantum mechanics in a nutshell. Inter- national Journal of Quantum Chemistry , 115(16):1058–1073, 2015

    2015

  4. [12]

    Machine-learning quantum states in the nisq era

    Giacomo Torlai and Roger G Melko. Machine-learning quantum states in the nisq era. Annual Review of Condensed Matter Physics , 11:325–344, 2020

    2020

  5. [13]

    A high-bias, low-variance introduction to machine learning for physicists

    Pankaj Mehta, Marin Bukov, Ching-Hao Wang, Alexandre GR Day, Clint Richardson, Charles K Fisher, and David J Schwab. A high-bias, low-variance introduction to machine learning for physicists. Physics reports, 810:1–124, 2019

    2019

  6. [14]

    The camels project: Cosmology and astrophysics with machine-learning simulations

    Francisco Villaescusa-Navarro, Daniel Angl´ es-Alc´ azar, Shy Genel, David N Spergel, Rachel S Somerville, Romeel Dave, Annalisa Pillepich, Lars Hern- 41 quist, Dylan Nelson, Paul Torrey, et al. The camels project: Cosmology and astrophysics with machine-learning simulations. ...

    2021

  7. [15]

    Machine learning and the physical sciences

    Giuseppe Carleo, Ignacio Cirac, Kyle Cranmer, Laurent Daudet, Maria Schuld, Naftali Tishby, Leslie Vogt-Maranto, and Lenka Zdeborov´ a. Machine learning and the physical sciences. Reviews of Modern Physics , 91(4):045002, 2019

    2019

  8. [16]

    Cosmoflow: Using deep learning to learn the universe at scale

    Amrita Mathuriya, Deborah Bard, Peter Mendygral, Lawrence Meadows, James Arnemann, Lei Shao, Siyu He, Tuomas K¨ arn¨ a, Diana Moise, Simon J Pennycook, et al. Cosmoflow: Using deep learning to learn the universe at scale. In SC18: International Conference for High Performance ...

    2018

  9. [17]

    Machine learning constraints on deviations from general relativity from the large scale structure of the universe

    George Alestas, Lavrentios Kazantzidis, and Savvas Nesseris. Machine learning constraints on deviations from general relativity from the large scale structure of the universe. Physical Review D , 106(10):103519, 2022

    2022

  10. [18]

    Methodus inveniendi lineas curvas maximi minimive propri- etate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti , volume 1

    Leonhard Euler. Methodus inveniendi lineas curvas maximi minimive propri- etate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti , volume 1. Springer Science & Business Media, 1952

    1952

  11. [19]

    ¨Uber die numerische aufl¨ osung von differentialgleichungen.Mathe- matische Annalen, 46(2):167–178, 1895

    Carl Runge. ¨Uber die numerische aufl¨ osung von differentialgleichungen.Mathe- matische Annalen, 46(2):167–178, 1895

  12. [20]

    Finite-element bending analysis for plates

    Leonard R Herrmann. Finite-element bending analysis for plates. Journal of the Engineering Mechanics Division , 93(5):13–26, 1967. 42

    1967

  13. [21]

    Lewis Fry Richardson. Ix. the approximate arithmetical solution by finite differ- ences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London. Series A, Containing ...

    1911

  14. [22]

    Finite volume meth- ods

    Robert Eymard, Thierry Gallou¨ et, and Rapha` ele Herbin. Finite volume meth- ods. Handbook of numerical analysis , 7:713–1018, 2000

    2000

  15. [23]

    Neural ordinary differential equations

    Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information process- ing systems , 31, 2018

    2018

  16. [24]

    Dgm: A deep learning algo- rithm for solving partial differential equations

    Justin Sirignano and Konstantinos Spiliopoulos. Dgm: A deep learning algo- rithm for solving partial differential equations. Journal of computational physics, 375:1339–1364, 2018

    2018

  17. [25]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Compu- tational physics , 378:686–707, 2019

    2019

  18. [26]

    Mri image reconstruction via learning optimization using neural odes

    Eric Z Chen, Terrence Chen, and Shanhui Sun. Mri image reconstruction via learning optimization using neural odes. In Medical Image Computing and Computer Assisted Intervention–MICCAI 2020: 23rd International Conference, Lima, Peru, October 4–8, 2020, Proceedings, Part II 23 ,...

    2020

  19. [27]

    Fore- casting the outcome of spintronic experiments with neural ordinary differential equations

    Xing Chen, Flavio Abreu Araujo, Mathieu Riou, Jacob Torrejon, Dafin´ e Rav- elosona, Wang Kang, Weisheng Zhao, Julie Grollier, and Damien Querlioz. Fore- casting the outcome of spintronic experiments with neural ordinary differential equations. Nature communications, 13(1):1016, 2022

    2022

  20. [28]

    Explainable tensorized neural ordinary differential equations for arbitrary-step time series prediction

    Penglei Gao, Xi Yang, Kaizhu Huang, Rui Zhang, and John Yannis Goulermas. Explainable tensorized neural ordinary differential equations for arbitrary-step time series prediction. IEEE Transactions on Knowledge and Data Engineering , 2022

    2022

  21. [29]

    The deep learning galerkin method for the general stokes equations

    Jian Li, Jing Yue, Wen Zhang, and Wansuo Duan. The deep learning galerkin method for the general stokes equations. Journal of Scientific Computing , 93(1):5, 2022

    2022

  22. [30]

    A deep learning galerkin method for the closed-loop geothermal system

    Wen Zhang and Jian Li. A deep learning galerkin method for the closed-loop geothermal system. arXiv preprint arXiv:2204.08139 , 2022

    2022 arXiv

  23. [31]

    Physics-informed neural networks (pinns) for fluid mechanics: A re- view

    Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Kar- niadakis. Physics-informed neural networks (pinns) for fluid mechanics: A re- view. Acta Mechanica Sinica, pages 1–12, 2022

    2022

  24. [32]

    Physics- informed neural networks for power systems

    George S Misyris, Andreas Venzke, and Spyros Chatzivasileiadis. Physics- informed neural networks for power systems. In 2020 IEEE Power & Energy Society General Meeting (PESGM) , pages 1–5. IEEE, 2020. 44

    2020

  25. [33]

    Physics-informed neural networks for heat transfer problems

    Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Kar- niadakis. Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer, 143(6), 2021

    2021

  26. [34]

    Stiff-pinn: Physics- informed neural network for stiff chemical kinetics

    Weiqi Ji, Weilun Qiu, Zhiyu Shi, Shaowu Pan, and Sili Deng. Stiff-pinn: Physics- informed neural network for stiff chemical kinetics. The Journal of Physical Chemistry A , 125(36):8098–8106, 2021

    2021

  27. [35]

    Towards efficient simulations of non-equilibrium chemistry in hypersonic flows: A physics-informed neural network framework

    Ivan Zanardi, Simone Venturi, and Marco Panesi. Towards efficient simulations of non-equilibrium chemistry in hypersonic flows: A physics-informed neural network framework. In AIAA SCITECH 2022 Forum , page 1639, 2022

    2022

  28. [36]

    A deep learning approach for predicting two-dimensional soil consolidation using physics-informed neural networks (pinn)

    Yue Lu and Gang Mei. A deep learning approach for predicting two-dimensional soil consolidation using physics-informed neural networks (pinn). Mathematics, 10(16):2949, 2022

    2022

  29. [37]

    A novel workflow based on physics-informed machine learning to deter- mine the permeability profile of fractured coal seams using downhole geophysical logs

    Adelina Lv, Lei Cheng, Mohammad Ali Aghighi, Hossein Masoumi, and Hamid Roshan. A novel workflow based on physics-informed machine learning to deter- mine the permeability profile of fractured coal seams using downhole geophysical logs. Marine and Petroleum Geology , 131:105171, 2021

    2021

  30. [38]

    Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

    Luning Sun, Han Gao, Shaowu Pan, and Jian-Xun Wang. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering , 361:112732, 2020

    2020

  31. [39]

    Deep learning of vortex-induced vibrations

    Maziar Raissi, Zhicheng Wang, Michael S Triantafyllou, and George Em Karni- 45 adakis. Deep learning of vortex-induced vibrations. Journal of Fluid Mechanics , 861:119–137, 2019

    2019

  32. [40]

    Nsfnets (navier- stokes flow nets): Physics-informed neural networks for the incompressible navier-stokes equations

    Xiaowei Jin, Shengze Cai, Hui Li, and George Em Karniadakis. Nsfnets (navier- stokes flow nets): Physics-informed neural networks for the incompressible navier-stokes equations. Journal of Computational Physics , 426:109951, 2021

    2021

  33. [41]

    Machine learning in cardiovascular flows modeling: Pre- dicting arterial blood pressure from non-invasive 4d flow mri data using physics- informed neural networks

    Georgios Kissas, Yibo Yang, Eileen Hwuang, Walter R Witschey, John A Detre, and Paris Perdikaris. Machine learning in cardiovascular flows modeling: Pre- dicting arterial blood pressure from non-invasive 4d flow mri data using physics- informed neural networks. Computer Method...

    2020

  34. [42]

    Physics-informed neural networks for cardiac activation mapping

    Francisco Sahli Costabal, Yibo Yang, Paris Perdikaris, Daniel E Hurtado, and Ellen Kuhl. Physics-informed neural networks for cardiac activation mapping. Frontiers in Physics, 8:42, 2020

    2020

  35. [43]

    Biologically-informed neural networks guide mechanis- tic modeling from sparse experimental data

    John H Lagergren, John T Nardini, Ruth E Baker, Matthew J Simpson, and Kevin B Flores. Biologically-informed neural networks guide mechanis- tic modeling from sparse experimental data. PLoS computational biology , 16(12):e1008462, 2020

    2020

  36. [44]

    Biologically informed neural networks predict drug responses

    Casey S Greene and James C Costello. Biologically informed neural networks predict drug responses. Cancer cell, 38(5):613–615, 2020

    2020

  37. [45]

    Systems biology informed deep learning for inferring parameters and hidden dynamics

    Alireza Yazdani, Lu Lu, Maziar Raissi, and George Em Karniadakis. Systems biology informed deep learning for inferring parameters and hidden dynamics. 46 PLoS computational biology, 16(11):e1007575, 2020

    2020

  38. [46]

    A generic physics-informed neural network-based constitutive model for soft biological tissues

    Minliang Liu, Liang Liang, and Wei Sun. A generic physics-informed neural network-based constitutive model for soft biological tissues. Computer methods in applied mechanics and engineering , 372:113402, 2020

    2020

  39. [47]

    Non-invasive inference of thrombus material properties with physics-informed neural networks

    Minglang Yin, Xiaoning Zheng, Jay D Humphrey, and George Em Karniadakis. Non-invasive inference of thrombus material properties with physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering , 375:113603, 2021

    2021

  40. [48]

    Characterizing possible failure modes in physics-informed neural networks

    Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-informed neural networks. Advances in Neural Information Processing Systems, 34:26548– 26560, 2021

    2021

  41. [49]

    Deepxde: A deep learning library for solving differential equations

    Lu Lu, Xuhui Meng, Zhiping Mao, and George Em Karniadakis. Deepxde: A deep learning library for solving differential equations. SIAM Review, 63(1):208– 228, 2021

    2021

  42. [50]

    When physics meets machine learning: A survey of physics-informed machine learning

    Chuizheng Meng, Sungyong Seo, Defu Cao, Sam Griesemer, and Yan Liu. When physics meets machine learning: A survey of physics-informed machine learning. arXiv preprint arXiv:2203.16797 , 2022

    2022 arXiv

  43. [51]

    Locally adap- tive activation functions with slope recovery for deep and physics-informed neural networks

    Ameya D Jagtap, Kenji Kawaguchi, and George Em Karniadakis. Locally adap- tive activation functions with slope recovery for deep and physics-informed neural networks. Proceedings of the Royal Society A , 476(2239):20200334, 2020. 47

    2020

  44. [52]

    Conserva- tive physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems

    Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conserva- tive physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering , 365:113028, 2020

    2020

  45. [53]

    Learning nonlinear operators via deeponet based on the universal ap- proximation theorem of operators

    Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karni- adakis. Learning nonlinear operators via deeponet based on the universal ap- proximation theorem of operators. Nature Machine Intelligence , 3(3):218–229, 2021

    2021

  46. [54]

    Distributed physics in- formed neural network for data-efficient solution to partial differential equations

    Vikas Dwivedi, Nishant Parashar, and Balaji Srinivasan. Distributed physics in- formed neural network for data-efficient solution to partial differential equations. arXiv preprint arXiv:1907.08967 , 2019

    1907 arXiv

  47. [55]

    Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations

    Ameya D Jagtap and George E Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. In AAAI Spring Symposium: MLPS , 2021

    2021

  48. [56]

    Gradient- enhanced physics-informed neural networks for forward and inverse pde prob- lems

    Jeremy Yu, Lu Lu, Xuhui Meng, and George Em Karniadakis. Gradient- enhanced physics-informed neural networks for forward and inverse pde prob- lems. Computer Methods in Applied Mechanics and Engineering , 393:114823, 2022

    2022

  49. [57]

    Deep learning-based method coupled with small sample learning for solving partial differential equations

    Ying Li and Fangjun Mei. Deep learning-based method coupled with small sample learning for solving partial differential equations. Multimedia Tools and 48 Applications, 80(11):17391–17413, 2021

    2021

  50. [58]

    Modified physics-informed neural network method based on the conservation law con- straint and its prediction of optical solitons

    Gang-Zhou Wu, Yin Fang, Yue-Yue Wang, and Chao-Qing Dai. Modified physics-informed neural network method based on the conservation law con- straint and its prediction of optical solitons. arXiv preprint arXiv:2108.13192 , 2021

    2021 arXiv

  51. [59]

    Physics-informed dynamic mode decomposition (pidmd)

    Peter J Baddoo, Benjamin Herrmann, Beverley J McKeon, J Nathan Kutz, and Steven L Brunton. Physics-informed dynamic mode decomposition (pidmd). arXiv preprint arXiv:2112.04307 , 2021

    2021 arXiv

  52. [60]

    Fourier neural networks

    Adrian Silvescu. Fourier neural networks. In IJCNN’99. International Joint Conference on Neural Networks. Proceedings (Cat. No. 99CH36339) , volume 1, pages 488–491. IEEE, 1999

    1999

  53. [61]

    Fourier neural operator for parametric partial differential equations

    Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08895 , 2020

    2010 arXiv

  54. [62]

    Towards large-scale learned solvers for parametric pdes with model-parallel fourier neural operators

    Thomas J Grady II, Rishi Khan, Mathias Louboutin, Ziyi Yin, Philipp A Witte, Ranveer Chandra, Russell J Hewett, and Felix J Herrmann. Towards large-scale learned solvers for parametric pdes with model-parallel fourier neural operators. arXiv preprint arXiv:2204.01205 , 2022

    2022 arXiv

  55. [63]

    Linear attention coupled fourier neural operator for simulation of three-dimensional turbulence

    Wenhui Peng, Zelong Yuan, Zhijie Li, and Jianchun Wang. Linear attention coupled fourier neural operator for simulation of three-dimensional turbulence. 49 arXiv preprint arXiv:2210.04259 , 2022

    2022 arXiv

  56. [64]

    Solving seismic wave equa- tions on variable velocity models with fourier neural operator

    Bian Li, Hanchen Wang, Xiu Yang, and Youzuo Lin. Solving seismic wave equa- tions on variable velocity models with fourier neural operator. arXiv preprint arXiv:2209.12340, 2022

    2022 arXiv

  57. [65]

    Learn- ing deep implicit fourier neural operators (ifnos) with applications to heteroge- neous material modeling

    Huaiqian You, Quinn Zhang, Colton J Ross, Chung-Hao Lee, and Yue Yu. Learn- ing deep implicit fourier neural operators (ifnos) with applications to heteroge- neous material modeling. arXiv preprint arXiv:2203.08205 , 2022

    2022 arXiv

  58. [66]

    Adaptive fourier neural operators: Efficient token mixers for transformers

    John Guibas, Morteza Mardani, Zongyi Li, Andrew Tao, Anima Anandkumar, and Bryan Catanzaro. Adaptive fourier neural operators: Efficient token mixers for transformers. arXiv preprint arXiv:2111.13587 , 2021

    2021 arXiv

  59. [67]

    A learning algo- rithm for boltzmann machines

    David H Ackley, Geoffrey E Hinton, and Terrence J Sejnowski. A learning algo- rithm for boltzmann machines. Cognitive science, 9(1):147–169, 1985

    1985

  60. [68]

    Rectified linear units improve restricted boltzmann machines

    Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Proceedings of the 27th international conference on machine learning (ICML-10) , pages 807–814, 2010

    2010

  61. [69]

    Deep sparse rectifier neural networks

    Xavier Glorot, Antoine Bordes, and Yoshua Bengio. Deep sparse rectifier neural networks. In Proceedings of the fourteenth international conference on artifi- cial intelligence and statistics , pages 315–323. JMLR Workshop and Conference Proceedings, 2011

    2011

  62. [70]

    Fast and ac- curate deep network learning by exponential linear units (elus)

    Djork-Arn´ e Clevert, Thomas Unterthiner, and Sepp Hochreiter. Fast and ac- curate deep network learning by exponential linear units (elus). arXiv preprint 50 arXiv:1511.07289, 2015

    2015 arXiv

  63. [71]

    Gaussian error linear units (gelus)

    Dan Hendrycks and Kevin Gimpel. Gaussian error linear units (gelus). arXiv preprint arXiv:1606.08415, 2016

    2016 arXiv

  64. [72]

    Implicit neural representations with periodic activation func- tions

    Vincent Sitzmann, Julien Martel, Alexander Bergman, David Lindell, and Gor- don Wetzstein. Implicit neural representations with periodic activation func- tions. Advances in Neural Information Processing Systems , 33:7462–7473, 2020

    2020

  65. [73]

    M. D. McKay, R. J. Beckman, and W. J. Conover. A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics, 21(2):239–245, 1979

    1979

  66. [74]

    A synthetic oscillatory network of transcriptional regulators

    Michael B Elowitz and Stanislas Leibler. A synthetic oscillatory network of transcriptional regulators. Nature, 403(6767):335–338, 2000

    2000

  67. [75]

    Discussion: the kermack-mckendrick epidemic threshold the- orem

    Roy M Anderson. Discussion: the kermack-mckendrick epidemic threshold the- orem. Bulletin of mathematical biology , 53(1):1–32, 1991

    1991

  68. [76]

    A modified age-structured sir model for covid-19 type viruses

    Vishaal Ram and Laura P Schaposnik. A modified age-structured sir model for covid-19 type viruses. Scientific reports, 11(1):1–15, 2021

    2021

  69. [77]

    The chemical basis of morphogenesis

    Alan Mathison Turing. The chemical basis of morphogenesis. Bulletin of math- ematical biology, 52(1):153–197, 1990

    1990

  70. [78]

    Turing’s model for biological pattern formation and the robustness problem

    Philip K Maini, Thomas E Woolley, Ruth E Baker, Eamonn A Gaffney, and S Seirin Lee. Turing’s model for biological pattern formation and the robustness problem. Interface focus, 2(4):487–496, 2012. 51 Appendix A: Systems Biology Models Systems biology models are computational t...

    2012

Pith tools

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