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arxiv: 2605.08408 · v1 · submitted 2026-05-08 · 💻 cs.LG

Recognition: no theorem link

AdamFLIP: Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN Training

Binghang Lu , Runyu Zhang , Changhong Mou , Na Li , Guang Lin
Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:47 UTC · model grok-4.3

classification 💻 cs.LG
keywords PINNconstrained optimizationfeedback linearizationAdam optimizerPDE solvinghard constraintsphysics-informed neural networksLagrangian optimization
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The pith

AdamFLIP reformulates PINN training as an equality-constrained optimization problem by using feedback linearization to drive constraint residuals into stable linear contraction dynamics before applying Adam adaptation to the Lagrangian.

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

The paper establishes a method to avoid the ill-conditioning and manual weight tuning of soft-penalty PINN training by treating the problem as hard equality-constrained optimization. It models the constraint residuals as the output of a controlled dynamical system and designs the Lagrange multiplier as a feedback input that forces those residuals to follow stable linear contraction. Adam-style first- and second-moment adaptation is then performed on the resulting linearized Lagrangian gradient. This combination is shown to deliver consistent gains on forward and inverse PDE benchmarks, with the largest reported improvement being a reduction of relative L2 error by more than two thirds on the Navier-Stokes test case relative to the next-best optimizer.

Core claim

AdamFLIP computes the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam-style first- and second-moment adaptation to the resulting feedback-linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural-network optimization.

What carries the argument

Feedback linearization of the constraint residual dynamics, which produces a controlled linear contraction that is then optimized by Adam on the Lagrangian gradient.

If this is right

  • The method yields lower solution errors than both standard soft-constrained PINNs and existing hard-constrained optimizers on forward and inverse PDE problems.
  • Hard constraint satisfaction is achieved without manual loss-weight balancing, reducing sensitivity to hyper-parameters.
  • The approach remains computationally scalable because it retains Adam-style moment adaptation on the modified gradient.
  • Improved satisfaction of initial and boundary conditions follows directly from driving residuals to contraction dynamics rather than penalizing them.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same feedback-linearization step could be applied to other equality-constrained neural-network training tasks outside PDEs, such as physics-informed generative models.
  • If the contraction rate can be adapted online, training speed on stiff PDEs might increase without sacrificing final accuracy.
  • The framework naturally supports mixed hard and soft constraints by treating only selected residuals with feedback linearization while leaving others in the loss.

Load-bearing premise

The feedback linearization produces a stable linear contraction dynamics for the constraint residuals that can be maintained throughout training while the Adam adaptation remains effective on the resulting Lagrangian gradient.

What would settle it

On the Navier-Stokes benchmark, failure to reduce relative L2 error by more than two thirds compared to the next best method, or observation that the constraint residuals do not contract according to the predicted linear dynamics during training.

Figures

Figures reproduced from arXiv: 2605.08408 by Binghang Lu, Changhong Mou, Guang Lin, Na Li, Runyu Zhang.

Figure 1
Figure 1. Figure 1: Illustration of the PINN framework. The network maps spatiotemporal inputs to the [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of spatiotemporal solutions and absolute errors for the one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of spatiotemporal solutions and absolute errors for the inverse one-dimensional [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Forward time-fractional mixed diffusion–wave equation (1D) results: benchmark spatiotem [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Inverse time-fractional mixed diffusion–wave equation (1D) results: spatiotemporal re [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of spatiotemporal solutions and absolute errors in forward problem for the 2d [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Inverse 2D heat equation results: comparison of absolute errors fields for each method at [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Pointwise absolute error |u − uˆ| (top row) and |v − vˆ| (bottom row) at t = 1 for the 2D Navier–Stokes equations. From left to right: Standard PINN, AL-PINN, trSQP-PINN, FL-PINN, and AdamFLIP. AdamFLIP produces the smallest and most spatially uniform errors across both velocity components. D Further Ablations [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

Physics-informed neural networks (PINNs) provide a flexible framework for solving forward and inverse problems governed by partial differential equations (PDEs), but standard PINN training typically relies on soft penalty formulations that combine PDE residuals, data mismatch, and initial/boundary conditions using manually chosen weights. This often leads to ill-conditioning, sensitivity to loss weights, and poor constraint satisfaction. In this work, we reformulate PINN training as an equality-constrained optimization problem and propose a novel Adaptive Momentum Feedback Linearization Optimization for Hard Constrained PINN (AdamFLIP). The key idea is to view the constraint residuals as the output of a controlled dynamical system and to compute the Lagrange multiplier as a feedback input that locally drives these residuals toward stable linear contraction dynamics. AdamFLIP then applies Adam-style first- and second-moment adaptation to the resulting feedback-linearized Lagrangian gradient, combining principled constraint handling with the scalability and robustness of adaptive neural-network optimization. We test AdamFLIP on a range of benchmark forward and inverse PDE problem, and it consistently outperforms both the standard soft-constrained PINN and state-of-the-art constrained optimizers. Specifically, on the Navier--Stokes equations benchmark, AdamFLIP \textbf{reduces relative $L_2$ error by more than two thirds} for the predicted solution compared to the next best method. Our AdamFLIP framework provides an effective and computationally scalable hard constraint optimization method for PINN training.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to introduce AdamFLIP for hard-constrained PINN training by reformulating it as equality-constrained optimization. It computes Lagrange multipliers using feedback linearization to drive constraint residuals to stable linear contraction dynamics, then optimizes the Lagrangian gradient with Adam. On benchmarks including Navier-Stokes, it outperforms soft PINNs and other methods, reducing relative L2 error by more than two thirds on Navier-Stokes.

Significance. If the contraction dynamics are preserved under adaptive updates, this provides a principled hard-constraint method combining feedback control with adaptive optimization, potentially improving PINN accuracy and robustness for PDEs. The reported error reduction on Navier-Stokes indicates significant practical impact if reproducible and if the stability assumption holds.

major comments (3)
  1. [Feedback linearization and Lagrangian gradient construction] The feedback linearization computes a Lagrange multiplier to enforce linear contraction on residuals, but the manuscript provides no analysis or verification showing that this closed-loop contraction persists when Adam updates alter the network parameters and thus the residual Jacobian (central to the hard-constraint claim).
  2. [Navier-Stokes experiments and results] The Navier-Stokes result (more than two-thirds relative L2 error reduction) is load-bearing for the performance claims, yet no metrics are reported on residual contraction rates or constraint violation norms during training to confirm the dynamics remain linear and contracting rather than reverting to an adaptive soft penalty.
  3. [Method hyperparameters and experimental setup] The feedback gain and contraction rate are free parameters whose selection is not analyzed for sensitivity; if chosen after seeing benchmark performance, this undermines the generality of the outperformance claims over soft-constrained PINNs and other optimizers.
minor comments (2)
  1. [Abstract] Abstract contains a grammatical error: 'a range of benchmark forward and inverse PDE problem' should read 'problems'.
  2. [Overall presentation] Notation for the feedback law, Lagrange multiplier, and Adam moment estimates could be clarified with explicit equations to distinguish the linearization step from the adaptive gradient update.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive comments on our manuscript. We address each of the major comments point by point below, providing clarifications and indicating the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: The feedback linearization computes a Lagrange multiplier to enforce linear contraction on residuals, but the manuscript provides no analysis or verification showing that this closed-loop contraction persists when Adam updates alter the network parameters and thus the residual Jacobian (central to the hard-constraint claim).

    Authors: We agree that the manuscript lacks a formal analysis proving that the contraction dynamics are preserved globally under the Adam updates, which modify the network parameters and thus the residual Jacobian at each step. The method recomputes the Lagrange multiplier at every iteration using the current Jacobian to enforce the local linear contraction, but the adaptive momentum in Adam introduces an approximation. This is a limitation of the current theoretical treatment. In the revised version, we will add a section discussing the local nature of the feedback linearization and include empirical evidence by reporting the evolution of the residual norms to show that the constraints remain well-satisfied throughout training. revision: partial

  2. Referee: The Navier-Stokes result (more than two-thirds relative L2 error reduction) is load-bearing for the performance claims, yet no metrics are reported on residual contraction rates or constraint violation norms during training to confirm the dynamics remain linear and contracting rather than reverting to an adaptive soft penalty.

    Authors: The referee correctly notes the absence of such metrics in the manuscript. To substantiate the hard-constraint enforcement, we will include additional plots in the revised manuscript showing the PDE residual norms and the achieved contraction rates over the course of training for the Navier-Stokes benchmark. These will demonstrate that the residuals are driven to zero in a manner consistent with the designed linear dynamics rather than behaving as a soft penalty. revision: yes

  3. Referee: The feedback gain and contraction rate are free parameters whose selection is not analyzed for sensitivity; if chosen after seeing benchmark performance, this undermines the generality of the outperformance claims over soft-constrained PINNs and other optimizers.

    Authors: We selected the feedback gain and contraction rate based on theoretical considerations for stability (ensuring the contraction rate is positive and the gain is sufficiently large) and preliminary experiments on simpler PDEs before evaluating on the full benchmarks. However, we acknowledge that a dedicated sensitivity analysis would strengthen the claims. In the revision, we will add a sensitivity study varying these parameters on the Navier-Stokes problem and report the resulting performance variations to confirm robustness. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation introduces independent feedback-linearized optimizer validated empirically

full rationale

The paper's central construction defines a feedback law that enforces linear contraction dynamics on residuals by design (standard feedback linearization), then applies Adam to the resulting Lagrangian gradient. This is not self-definitional or a fitted input renamed as prediction; the contraction property follows directly from the chosen feedback gain and is not claimed as an emergent result. Performance claims rest on benchmark comparisons rather than any self-citation chain or uniqueness theorem imported from prior author work. No load-bearing step reduces to its own inputs by construction, and the method remains falsifiable via external PDE benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The method rests on the assumption that the constraint residuals can be treated as the output of a controlled dynamical system whose local linearization admits a stabilizing feedback law that remains valid during neural network training. It also assumes that the resulting constrained gradient can be safely adapted with first- and second-moment estimates without destroying the contraction property.

free parameters (2)
  • feedback gain / contraction rate
    The rate at which residuals are driven to zero is a design parameter that must be chosen or adapted; its value directly affects whether the linear contraction holds.
  • Adam hyperparameters (beta1, beta2, epsilon)
    Standard Adam parameters are used on the feedback-linearized gradient; their specific values are not derived from first principles.
axioms (2)
  • domain assumption The map from network parameters to PDE residuals admits a locally linearizable controlled dynamics
    Invoked when the method treats residuals as the state of a dynamical system that can be feedback-linearized.
  • domain assumption The Lagrange multiplier computed via feedback linearization yields a gradient that can be safely adapted by Adam without violating constraint satisfaction
    Central to combining the control law with adaptive optimization.

pith-pipeline@v0.9.0 · 5569 in / 1522 out tokens · 38560 ms · 2026-05-12T00:47:44.828556+00:00 · methodology

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Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages · 1 internal anchor

  1. [1]

    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 Computational physics, 378:686–707, 2019

    work page 2019

  2. [2]

    Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    work page 2021

  3. [3]

    Scientific machine learning through physics–informed neural networks: Where we are and what’s next.Journal of Scientific Computing, 92(3):88, 2022

    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(3):88, 2022

    work page 2022

  4. [4]

    Physics-informed neural networks (pinns) for fluid mechanics: A review.Acta Mechanica Sinica, 37(12):1727–1738, 2021

    Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis. Physics-informed neural networks (pinns) for fluid mechanics: A review.Acta Mechanica Sinica, 37(12):1727–1738, 2021

    work page 2021

  5. [5]

    Jiangtao Guo, Hao Zhu, Yujie Yang, and Chenrui Guo. Advances in physics-informed neural networks for solving complex partial differential equations and their engineering applications: A systematic review.Engineering Applications of Artificial Intelligence, 161:112044, 2025

    work page 2025

  6. [6]

    Understanding and mitigating gradient flow pathologies in physics-informed neural networks.SIAM Journal on Scientific Computing, 43(5):A3055–A3081, 2021

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

    work page 2021

  7. [7]

    Characterizing possible failure modes in physics-informed neural networks.Advances in neural information processing systems, 34:26548–26560, 2021

    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

    work page 2021

  8. [8]

    An expert’s guide to training physics-informed neural networks.arXiv preprint arXiv:2308.08468, 2023

    Sifan Wang, Shyam Sankaran, Hanwen Wang, and Paris Perdikaris. An expert’s guide to training physics-informed neural networks.arXiv preprint arXiv:2308.08468, 2023

    work page arXiv 2023

  9. [9]

    Inverse dirichlet weighting enables reliable training of physics informed neural networks.Machine Learning: Science and Technology, 3(1):015026, 2022

    Suryanarayana Maddu, Dominik Sturm, Christian L Müller, and Ivo F Sbalzarini. Inverse dirichlet weighting enables reliable training of physics informed neural networks.Machine Learning: Science and Technology, 3(1):015026, 2022

    work page 2022

  10. [10]

    Trust region method for coupled systems of pde solvers and deep neural networks.arXiv preprint arXiv:2105.07552, 2021

    Kailai Xu and Eric Darve. Trust region method for coupled systems of pde solvers and deep neural networks.arXiv preprint arXiv:2105.07552, 2021

    work page arXiv 2021

  11. [11]

    The admm-pinns algorithmic framework for nonsmooth pde-constrained optimization: A deep learning approach.SIAM Journal on Scientific Computing, 46(6):C659–C687, 2024

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue. The admm-pinns algorithmic framework for nonsmooth pde-constrained optimization: A deep learning approach.SIAM Journal on Scientific Computing, 46(6):C659–C687, 2024

    work page 2024

  12. [12]

    Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G. Johnson. Physics-informed neural networks with hard constraints for inverse design.arXiv preprint arXiv:2102.04626, 2021

    work page arXiv 2021

  13. [13]

    Enhanced physics-informed neural net- works with augmented lagrangian relaxation method (al-pinns).Neurocomputing, 548:126424, 2023

    Hwijae Son, Sung Woong Cho, and Hyung Ju Hwang. Enhanced physics-informed neural net- works with augmented lagrangian relaxation method (al-pinns).Neurocomputing, 548:126424, 2023

    work page 2023

  14. [14]

    Conditionally adaptive augmented la- grangian method for physics-informed learning of forward and inverse problems.arXiv preprint arXiv:2508.15695, 2025

    Qifeng Hu, Shamsulhaq Basir, and Inanc Senocak. Conditionally adaptive augmented la- grangian method for physics-informed learning of forward and inverse problems.arXiv preprint arXiv:2508.15695, 2025. 10

    work page arXiv 2025

  15. [15]

    Physics-informed neural networks with trust-region sequential quadratic programming.arXiv preprint arXiv:2409.10777, 2024

    Xiaoran Cheng and Sen Na. Physics-informed neural networks with trust-region sequential quadratic programming.arXiv preprint arXiv:2409.10777, 2024

    work page arXiv 2024

  16. [16]

    Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5):987– 1000, 1998

    Isaac E Lagaris, Aristidis Likas, and Dimitrios I Fotiadis. Artificial neural networks for solving ordinary and partial differential equations.IEEE transactions on neural networks, 9(5):987– 1000, 1998

    work page 1998

  17. [17]

    Neural networks with physics-informed architectures and constraints for dynamical systems modeling

    Franck Djeumou, Cyrus Neary, Eric Goubault, Sylvie Putot, and Ufuk Topcu. Neural networks with physics-informed architectures and constraints for dynamical systems modeling. In Learning for Dynamics and Control Conference, pages 263–277. PMLR, 2022

    work page 2022

  18. [18]

    Hard-constraining neumann boundary conditions in physics-informed neural networks via fourier feature embed- dings.arXiv preprint arXiv:2504.01093, 2025

    Christopher Straub, Philipp Brendel, Vlad Medvedev, and Andreas Rosskopf. Hard-constraining neumann boundary conditions in physics-informed neural networks via fourier feature embed- dings.arXiv preprint arXiv:2504.01093, 2025

    work page arXiv 2025

  19. [19]

    Physical informed neural networks with soft and hard boundary constraints for solving advection- diffusion equations using fourier expansions.Computers & Mathematics with Applications, 159:60–75, 2024

    work page 2024

  20. [20]

    Physics-informed neural networks with hard linear equality constraints.Computers & Chemical Engineering, 189:108764, 2024

    Hao Chen, Gonzalo E Constante Flores, and Can Li. Physics-informed neural networks with hard linear equality constraints.Computers & Chemical Engineering, 189:108764, 2024

    work page 2024

  21. [21]

    The Sample Average Approximation Method for Stochastic Discrete Optimization,

    James Kotary, Ferdinando Fioretto, Pascal Van Hentenryck, and Bryan Wilder. End-to-end constrained optimization learning: A survey.arXiv preprint arXiv:2103.16378, 2021

    work page arXiv 2021

  22. [22]

    A comprehensive survey on safe reinforcement learning

    Javier Garcıa and Fernando Fernández. A comprehensive survey on safe reinforcement learning. Journal of Machine Learning Research, 16(1):1437–1480, 2015

    work page 2015

  23. [23]

    A primal dual formulation for deep learning with constraints.Advances in neural information processing systems, 32, 2019

    Yatin Nandwani, Abhishek Pathak, and Parag Singla. A primal dual formulation for deep learning with constraints.Advances in neural information processing systems, 32, 2019

    work page 2019

  24. [24]

    Responsive safety in reinforcement learning by pid lagrangian methods

    Adam Stooke, Joshua Achiam, and Pieter Abbeel. Responsive safety in reinforcement learning by pid lagrangian methods. InInternational conference on machine learning, pages 9133–9143. PMLR, 2020

    work page 2020

  25. [25]

    Sequential quadratic optimization for nonlinear equality constrained stochastic optimization.SIAM Journal on Optimization, 31(2):1352–1379, 2021

    Albert S Berahas, Frank E Curtis, Daniel Robinson, and Baoyu Zhou. Sequential quadratic optimization for nonlinear equality constrained stochastic optimization.SIAM Journal on Optimization, 31(2):1352–1379, 2021

    work page 2021

  26. [26]

    Hardnet: Hard-constrained neural networks with universal approximation guarantees.arXiv preprint arXiv:2410.10807,

    Youngjae Min and Navid Azizan. Hardnet: Hard-constrained neural networks with universal approximation guarantees.arXiv preprint arXiv:2410.10807, 2024

    work page arXiv 2024

  27. [27]

    Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers.arXiv preprint arXiv:2508.10480, 2025

    Panagiotis D Grontas, Antonio Terpin, Efe C Balta, Raffaello D’Andrea, and John Lygeros. Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers.arXiv preprint arXiv:2508.10480, 2025

    work page arXiv 2025

  28. [28]

    Constrained optimization from a control perspective via feedback linearization

    Runyu Zhang, Arvind Raghunathan, Jeff S Shamma, and Na Li. Constrained optimization from a control perspective via feedback linearization. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2025

    work page 2025

  29. [29]

    Zeroth-order constrained optimization from a control perspective via feedback linearization.arXiv preprint arXiv:2509.24056, 2025

    Runyu Zhang, Gioele Zardini, Asuman Ozdaglar, Jeff Shamma, and Na Li. Zeroth-order constrained optimization from a control perspective via feedback linearization.arXiv preprint arXiv:2509.24056, 2025

    work page arXiv 2025

  30. [30]

    A new framework for constrained optimization via feedback control of lagrange multipliers.IEEE Transactions on Automatic Control, 2025

    Vito Cerone, Sophie M Fosson, Simone Pirrera, and Diego Regruto. A new framework for constrained optimization via feedback control of lagrange multipliers.IEEE Transactions on Automatic Control, 2025

    work page 2025

  31. [31]

    Adam: A Method for Stochastic Optimization

    Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization.arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    Dynamic weight strategy of physics-informed neural networks for the 2d navier–stokes equations.Entropy, 24(9):1254, 2022

    Shirong Li and Xinlong Feng. Dynamic weight strategy of physics-informed neural networks for the 2d navier–stokes equations.Entropy, 24(9):1254, 2022

    work page 2022

  33. [33]

    Rathore, W

    Pratik Rathore, Weimu Lei, Zachary Frangella, Lu Lu, and Madeleine Udell. Challenges in training pinns: A loss landscape perspective.arXiv preprint arXiv:2402.01868, 2024. 11

    work page arXiv 2024

  34. [34]

    Understanding the difficulty of solving cauchy problems with pinns

    Tao Wang, Bo Zhao, Sicun Gao, and Rose Yu. Understanding the difficulty of solving cauchy problems with pinns. In6th Annual Learning for Dynamics & Control Conference, pages 453–465. PMLR, 2024

    work page 2024

  35. [35]

    Error estimates for physics-informed neural networks approximating the navier–stokes equations.IMA Journal of Numerical Analysis, 44(1):83–119, 2024

    Tim De Ryck, Ameya D Jagtap, and Siddhartha Mishra. Error estimates for physics-informed neural networks approximating the navier–stokes equations.IMA Journal of Numerical Analysis, 44(1):83–119, 2024

    work page 2024

  36. [36]

    On the convergence of adam and beyond

    Sashank J Reddi, Satyen Kale, and Sanjiv Kumar. On the convergence of adam and beyond. In International Conference on Learning Representations, 2018

    work page 2018

  37. [37]

    Springer, 2006

    Jorge Nocedal and Stephen J Wright.Numerical optimization. Springer, 2006

    work page 2006

  38. [38]

    A review of constraint qualifications in finite-dimensional spaces.Siam Review, 15(3):639–654, 1973

    David W Peterson. A review of constraint qualifications in finite-dimensional spaces.Siam Review, 15(3):639–654, 1973

    work page 1973

  39. [39]

    Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation.Journal of Mathematical Analysis and Applications, 374(2):538–548, 2011

    Yuri Luchko. Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation.Journal of Mathematical Analysis and Applications, 374(2):538–548, 2011

    work page 2011

  40. [40]

    Lie point symmetry data aug- mentation for neural PDE solvers

    Johannes Brandstetter, Max Welling, and Daniel E Worrall. Lie point symmetry data aug- mentation for neural PDE solvers. InInternational Conference on Machine Learning, pages 2241–2256. PMLR, 2022

    work page 2022

  41. [41]

    Akhound-Sadegh, L

    T. Akhound-Sadegh, L. Perreault-Levasseur, J. Brandstetter, M. Welling, and S. Ravanbakhsh. Lie point symmetry and physics-informed networks. InAdvances in Neural Information Processing Systems, volume 36, pages 42468–42481, 2023

    work page 2023

  42. [42]

    Mialon, Q

    G. Mialon, Q. Garrido, H. Lawrence, D. Rehman, Y . LeCun, and B. Kiani. Self-supervised learning with lie symmetries for partial differential equations. InAdvances in Neural Information Processing Systems, volume 36, pages 28973–29004, 2023

    work page 2023

  43. [43]

    Physics-constrained flow matching: Sampling generative models with hard constraints.arXiv preprint arXiv:2506.04171, 2025

    Utkarsh Utkarsh, Pengfei Cai, Alan Edelman, Rafael Gomez-Bombarelli, and Christopher Vin- cent Rackauckas. Physics-constrained flow matching: Sampling generative models with hard constraints.arXiv preprint arXiv:2506.04171, 2025. A Related Works: Constrained Optimization Methods for PINN, and comparison with AdamFLIP In this section, we review recent cons...

    work page arXiv 2025

  44. [44]

    is a coordinatewise convex combination of g◦2 1 , . . . , g◦2 t . Hence, by Assumption 1, for every coordinatej, 0≤˜vt,j ≤G 2. The monotone second-moment estimate is defined by ¯vt = max elem {¯vt−1,˜vt}. Thus0≤¯v t,j ≤G 2 for allt, j. Since Dt = diag 1√¯vt−1 +δ , 30 we have, for every coordinatej, 1 G+δ ≤ 1√¯vt−1,j +δ ≤ 1 δ . This proves the uniform metr...

    work page