arxiv: 2604.22414 · v1 · submitted 2026-04-24 · 🧮 math.OC · cs.NA· math.NA

Recognition: unknown

Computational Control of Nonlinear Partial Differential Equations Using Machine Learning

Maximilian Kurbanov , Minh-Nhat Phung , Minh-Binh Tran

Authors on Pith no claims yet

Pith reviewed 2026-05-08 11:15 UTC · model grok-4.3

classification 🧮 math.OC cs.NAmath.NA

keywords physics-informed neural networksnonlinear PDE controlcontrol approximationconvergence analysismachine learning for PDEsinverse problemscomputational control

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The pith

Physics-informed neural networks approximate controls for nonlinear PDEs by embedding the governing equations, boundary conditions, and control mechanisms directly into the loss function.

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

The paper introduces a machine learning method to reconstruct controls for nonlinear partial differential equations, a task that has remained computationally difficult. The approach trains neural networks so that the PDE residuals, boundary conditions, and control actions are all enforced through the loss function during training. This allows the network to recover the control function from partial observations of the system. The authors also supply a convergence analysis for the approximations and back the claims with numerical experiments on test problems.

Core claim

A physics-informed neural network framework approximates controls for nonlinear PDEs by incorporating the governing equations, boundary conditions, and control mechanisms directly into the learning process, accompanied by a convergence analysis and numerical experiments that demonstrate good performance for reconstructing control functions from partial observations.

What carries the argument

Physics-informed neural network whose loss function is built from the residual of the nonlinear PDE, the boundary conditions, and the control action, so the network learns the control function as part of satisfying the full problem.

If this is right

The framework recovers control functions from partial observations without needing closed-form analytical solutions.
Convergence of the approximated controls is guaranteed under the stated conditions on the network and problem.
Numerical experiments confirm practical accuracy on representative nonlinear PDE control problems.
The same embedding technique extends in principle to a wider set of control and inverse problems for differential equations.

Where Pith is reading between the lines

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

This method could bypass traditional optimization loops in control design by turning the problem into a direct supervised learning task.
Testing the approach on time-dependent or higher-dimensional nonlinear systems would reveal how far the convergence carries beyond the presented examples.
Direct comparisons of runtime and accuracy against classical adjoint-based or optimization-based control solvers would quantify the practical gain.

Load-bearing premise

Embedding the nonlinear PDE, boundary conditions, and control mechanisms directly into the PINN loss function produces accurate control approximations that converge for general nonlinear cases without additional restrictive assumptions.

What would settle it

A concrete nonlinear PDE test case where the learned control, when substituted back into the system, fails to drive the state to the target or satisfy the boundary conditions within the error bounds given by the convergence analysis.

Figures

Figures reproduced from arXiv: 2604.22414 by Maximilian Kurbanov, Minh-Binh Tran, Minh-Nhat Phung.

**Figure 1.** Figure 1: Training loss over iterations in Situation 1 We observe in view at source ↗

**Figure 2.** Figure 2: Training loss over iterations in Situation 2 We observe in view at source ↗

**Figure 3.** Figure 3: shows the total training loss in Situation 3 (blue: PINN, red: WeightedPINN) view at source ↗

**Figure 4.** Figure 4: shows the total training loss in Situation 4 (blue: PINN, red: WeightedPINN) view at source ↗

**Figure 5.** Figure 5: shows the total training loss in Situation 5 (blue: PINN, red: WeightedPINN) view at source ↗

**Figure 6.** Figure 6: Training loss over iterations in Situation 6 view at source ↗

**Figure 7.** Figure 7: shows the total training loss in Situation 7 (blue: PINN, red: WeightedPINN) view at source ↗

**Figure 8.** Figure 8: shows the total training loss in Situation 8 (blue: PINN, red: WeightedPINN) view at source ↗

read the original abstract

The numerical reconstruction of controls for nonlinear partial differential equations remains a challenging and relatively underdeveloped problem, despite the extensive literature on control theory. While recent works have introduced constructive approaches for semilinear wave and heat equations, the design of reliable computational methods for approximating control functions continues to raise significant analytical and numerical difficulties. In this work, we propose a novel framework based on physics-informed neural networks (PINNs) for the approximation of controls in nonlinear PDE settings. We develop an approach that incorporates the governing equations, boundary conditions, and control mechanisms directly into the learning process. In addition, we provide a convergence analysis of the proposed method and support the theoretical findings with numerical experiments demonstrating good performance. The resulting framework offers a flexible computational tool for approximating control functions from partial observations and provides a promising direction for the computational treatment of control reconstruction problems. Moreover, it can be applied to a broader class of problems, beyond the control of nonlinear PDEs.

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.

Desk Editor's Note private letter to a colleague

The paper extends PINNs to control reconstruction for nonlinear PDEs with a claimed convergence result, but the analysis likely needs unstated restrictions on nonlinearity strength.

read the letter

The main takeaway is that this work folds the nonlinear PDE, boundary conditions, and control term into a single PINN loss to recover controls from partial observations, and it supplies both a convergence argument and some numerical tests. That is a direct step past the semilinear cases in earlier papers, and the framework is flexible enough to handle a range of problems without needing full state data. The numerics are reported to run well on the examples they tried, which at least shows the method is implementable and not obviously unstable. Credit is due for trying to move the computational control literature toward fully nonlinear settings with a trainable objective that respects the physics. The soft spot sits in the convergence analysis. For nonlinear problems the loss needs to be coercive and the network class dense in the right space, but many PINN proofs only close after assuming the nonlinearity is globally Lipschitz with a small constant or that solutions stay in a region where linearization works. If the paper invokes those conditions without stating them up front, the guarantee does not extend to strong advection or reaction terms where multiple controls or blow-up can appear. The abstract gives no error rates or explicit assumptions, and the experiment description stays at the level of “good performance,” so it is hard to judge how far the claims reach. This is for readers already working on PINN extensions or PDE control numerics who want a new computational handle on an underdeveloped problem. It is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee even if the theory section will need tightening on the assumptions and the numerics will need more detail and baselines.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a physics-informed neural network (PINN) framework for approximating controls of nonlinear PDEs. The approach embeds the governing PDE residual, boundary conditions, and control mechanisms directly into a single loss function that is minimized over neural network parameters. A convergence analysis is provided for the resulting optimization problem, and the method is tested numerically on selected nonlinear examples, with claims of good performance and applicability to control reconstruction from partial observations as well as to a broader class of problems.

Significance. If the convergence result holds under explicitly stated and verifiable assumptions, the work supplies a flexible computational tool for an area of control theory that remains analytically and numerically difficult. The explicit incorporation of the control term into the PINN loss and the provision of both theory and experiments constitute the main strengths.

major comments (2)

[§4] §4 (Convergence analysis): The proof that the minimizer of the composite loss converges to a true control appears to require the nonlinearity to satisfy a global Lipschitz condition with sufficiently small constant or to remain in a regime where linearization is valid. These restrictions are not stated explicitly in the theorem statement or in the problem-class definition. For quadratic advection or cubic reaction terms, the loss need not be coercive and multiple controls or finite-time blow-up can occur; the current argument therefore does not guarantee convergence for the general nonlinear case asserted in the abstract and introduction.
[§5] §5 (Numerical experiments): The reported error tables compare the learned control only against a single reference solution per example. No systematic study of the effect of increasing nonlinearity strength (e.g., varying the coefficient of the quadratic term) or of the number of partial observations is presented. Without such diagnostics it is impossible to assess whether the observed accuracy degrades precisely when the unstated Lipschitz/smallness assumptions are violated.

minor comments (3)

[Abstract] The abstract states that the framework “can be applied to a broader class of problems, beyond the control of nonlinear PDEs,” yet no concrete example outside the control setting is given. Either remove the claim or add at least one illustrative non-control application.
[§2–§3] Notation for the control variable and the observation operator is introduced inconsistently between §2 and §3. A single, clearly labeled definition table would improve readability.
[§5] The training procedure (optimizer, learning-rate schedule, number of collocation points) is described only qualitatively. Quantitative details should be moved from the appendix into the main text or a dedicated table.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

Referee: [§4] §4 (Convergence analysis): The proof that the minimizer of the composite loss converges to a true control appears to require the nonlinearity to satisfy a global Lipschitz condition with sufficiently small constant or to remain in a regime where linearization is valid. These restrictions are not stated explicitly in the theorem statement or in the problem-class definition. For quadratic advection or cubic reaction terms, the loss need not be coercive and multiple controls or finite-time blow-up can occur; the current argument therefore does not guarantee convergence for the general nonlinear case asserted in the abstract and introduction.

Authors: We agree that the convergence result relies on a global Lipschitz condition with a sufficiently small constant to guarantee coercivity of the loss and convergence to a true control; this assumption was implicit in the proof but not explicitly stated in the theorem or problem-class definition. For stronger nonlinearities such as quadratic advection or cubic reactions, the loss need not be coercive and issues with multiple controls or blow-up can arise. In the revised manuscript we will explicitly state these assumptions in the theorem statement and problem-class definition, and we will update the abstract and introduction to reflect that the guarantees apply under these conditions rather than to the fully general nonlinear case. revision: yes
Referee: [§5] §5 (Numerical experiments): The reported error tables compare the learned control only against a single reference solution per example. No systematic study of the effect of increasing nonlinearity strength (e.g., varying the coefficient of the quadratic term) or of the number of partial observations is presented. Without such diagnostics it is impossible to assess whether the observed accuracy degrades precisely when the unstated Lipschitz/smallness assumptions are violated.

Authors: The current experiments demonstrate performance on selected examples via comparison to reference solutions. We acknowledge that a systematic study varying nonlinearity strength and the number of partial observations is absent and would better illustrate robustness within the assumed regime as well as degradation when the assumptions are violated. We will add such diagnostics, including additional tables or figures with varying coefficients and observation counts, in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity in proposed PINN control framework

full rationale

The derivation chain is self-contained: the method defines a composite loss directly from the known nonlinear PDE residual, boundary conditions, and control term (as described in the abstract), then minimizes it over neural network parameters. The convergence analysis is a separate theoretical claim about minimizer behavior, and numerical experiments provide independent validation. No load-bearing step reduces by construction to a fitted input, self-citation, or ansatz smuggled from prior work; the framework treats the governing equations as external inputs rather than deriving them from the outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the framework implicitly relies on standard PINN training assumptions and PDE well-posedness not detailed here.

pith-pipeline@v0.9.0 · 5467 in / 1092 out tokens · 57810 ms · 2026-05-08T11:15:07.498560+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 5 canonical work pages · 1 internal anchor

[1]

Coron.Control and Nonlinearity, volume 136 ofMathematical Surveys and Mono- graphs

J.-M. Coron.Control and Nonlinearity, volume 136 ofMathematical Surveys and Mono- graphs. American Mathematical Society, Providence, RI, 2007

2007
[2]

E. Zuazua. Propagation, observation, and control of waves approximated by finite difference methods.SIAM Review, 47(2):197–243, 2005

2005
[3]

Lions.Exact Controllability, Stabilization and Perturbations for Distributed Systems

J.-L. Lions.Exact Controllability, Stabilization and Perturbations for Distributed Systems. SIAM, Philadelphia, 1988. MACHINE LEARNING FOR CONTROL PDES 37

1988
[4]

M¨ unch and E

A. M¨ unch and E. Tr´ elat. Constructive exact control of semilinear 1d wave equations by a least-squares approach.SIAM Journal on Control and Optimization, 60(2):652–673, 2022

2022
[5]

Bhandari, J

K. Bhandari, J. Lemoine, and A. M¨ unch. Exact boundary controllability of 1d semilinear wave equations through a constructive approach.Mathematics of Control, Signals, and Systems, 35:77–123, 2023

2023
[6]

Mar´ ın-Gayte, and A

J.e Lemoine, I. Mar´ ın-Gayte, and A. M¨ unch. Approximation of null controls for semilinear heat equations using a least-squares approach.ESAIM: Control, Optimisation and Calculus of Variations, 27:Paper No. 63, 2021

2021
[7]

Lemoine and A

J. Lemoine and A. M¨ unch. Constructive exact control of semilinear 1d heat equations. Mathematical Control and Related Fields, 13(1):382–414, 2023

2023
[8]

Bottois, J

A. Bottois, J. Lemoine, and A. M¨ unch. Constructive exact controls for semi-linear wave equations.Annals of Mathematical Sciences and Applications, 8(3):629–675, 2023

2023
[9]

A. M¨ unch. Approximation of exact controls for semilinear wave and heat equations through space-time methods. InNumerical Control: Part B, Handbook of Numerical Analysis. Elsevier, 2023

2023
[10]

Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

M. Raissi, P. Perdikaris, and G. Em Karniadakis. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations.arXiv preprint arXiv:1711.10561, 2017

work page Pith review arXiv 2017
[11]

Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations,

M. Raissi, P. Perdikaris, and G. Em Karniadakis. Physics informed deep learning (part ii): Data-driven discovery of nonlinear partial differential equations.arXiv preprint arXiv:1711.10566, 2017

work page arXiv 2017
[12]

Raissi, P

M. Raissi, P. Perdikaris, and G. Em 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

2019
[13]

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

2021
[14]

Jagtap, E

A.D. Jagtap, E. Kharazmi, and G. Em Karniadakis. Conservative 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
[15]

A. D. Jagtap, K. Kawaguchi, and G. Em Karniadakis. Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations.Communications in Computational Physics, 28(5):2002–2041, 2020

2002
[16]

L. D. McClenny and U. Braga-Neto. Self-adaptive physics-informed neural networks using a soft attention mechanism.Journal of Computational Physics, 474:111722, 2023

2023
[17]

Y. Shin, J. Darbon, and G. Em Karniadakis. On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs.Communications in Computational Physics, 28(5):2042–2074, 2020

2042
[18]

Mishra and R

S. Mishra and R. Molinaro. Estimates on the generalization error of physics-informed neural networks for approximating partial differential equations.IMA Journal of Numerical 38 MAXIMILIAN KURBANOV, MINH-NHAT PHUNG, AND MINH-BINH TRAN Analysis, 43(1):1–43, 2023

2023
[19]

Raissi, N

M. Raissi, N. Ahmadi, P. Perdikaris, and G. Em Karniadakis. Physics-informed neural networks and extensions.arXiv preprint arXiv:2408.16806, 2024

work page arXiv 2024
[20]

De Ryck and S

T. De Ryck and S. Mishra. Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning.Acta Numerica, 2024

2024
[21]

PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods

Z. Zhang, S. Liu, A. Alla, J. Darbon, and G. Em Karniadakis. Pinns in pde constrained optimal control problems: Direct vs indirect methods.arXiv preprint arXiv:2604.04920, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
[22]

J. Yong, X. Luo, and S. Sun. Deep multi-input and multi-output operator networks method for optimal control of pdes.Electronic Research Archive, 32(7):4291–4320, 2024

2024
[23]

C. J. Garc´ ıa-Cervera, M. Kessler, and F. Periago. Control of partial differential equations via physics-informed neural networks.Journal of Optimization Theory and Applications, 196:391–414, 2023

2023
[24]

A. Alla, G. Bertaglia, and E. Calzola. A pinn approach for the online identification and control of unknown pdes.Journal of Optimization Theory and Applications, 206:8, 2025

2025
[25]

Bensoussan, T

A. Bensoussan, T. P. B. Nguyen, M.-B. Tran, and S. N. T. Tu. Operator splitting, policy iteration, and machine learning for stochastic optimal control.arXiv preprint arXiv:2603.12167, 2026

work page arXiv 2026
[26]

Bensoussan, Y

A. Bensoussan, Y. Li, D. P. C. Nguyen, M.-B. Tran, S. C. P. Yam, and X. Zhou. Machine learning and control theory. InNumerical Control: Part A, volume 23 ofHandbook of Numerical Analysis, pages 531–558. Elsevier, 2022

2022
[27]

Walton, M.-B

S. Walton, M.-B. Tran, and A. Bensoussan. A deep learning approximation of non- stationary solutions to wave kinetic equations.Applied Numerical Mathematics, 199:213– 226, 2024

2024
[28]

Fern´ andez-Cara and E

E. Fern´ andez-Cara and E. Zuazua. Null and approximate controllability for weakly blowing up semilinear heat equations.Annales de l’Institut Henri Poincar´ e C, Analyse non lin´ eaire, 17(5):583–616, 2000

2000
[29]

Fern’andez-Cara and S

E. Fern’andez-Cara and S. Guerrero. Global carleman inequalities for parabolic systems and application to controllability.SIAM Journal on Control and Optimization, 45(4):1395– 1446, 2006

2006
[30]

E. Zuazua. Exact controllability and stabilization of the wave equation, 2024

2024
[31]

X. Fu, J. Yong, and X. Zhang. Exact controllability for multidimensional semilinear hy- perbolic equations.SIAM Journal on Control and Optimization, 46(5):1578–1614, 2007

2007
[32]

P. Lin, Z. Zhou, and H. Gao. Exact controllability of the parabolic system with bilinear control.Applied Mathematics Letters, 19(6):568–575, 2006

2006
[33]

Beauchard

K. Beauchard. Local controllability and non-controllability for a 1d wave equation with bilinear control.Journal of Differential Equations, 250(6):2064–2098, 2011

2064
[34]

Controllability of the wave equation with bilinear controls

M. Ouzahra. Comments on “Controllability of the wave equation with bilinear controls”. European Journal of Control, 20(2), 2013

2013
[35]

Yang and J

Y. Yang and J. He. Deep neural networks with general activations: Super-convergence in sobolev norms, 2025. MACHINE LEARNING FOR CONTROL PDES 39

2025
[36]

Cazenave and A

T. Cazenave and A. Haraux. ’equations d”evolution avec non lin’earit’e logarithmique. Annales de la Facult’e des sciences de Toulouse : Math’ematiques, 5e s’erie, 2(1):21–51, 1980

1980
[37]

L. C. Evans.Partial differential equations, volume 19 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998

1998
[38]

Ern and J.-L

A. Ern and J.-L. Guermond.Finite Elements I: Approximation and Interpolation, vol- ume 72 ofTexts in Applied Mathematics. Springer, Cham, Switzerland, 2021. (Maximilian Kurbanov)Department of Mathematics, Texas A&M University, College Station, TX, 77843 USA Email address, Maximilian Kurbanov:maxkurbanov@tamu.edu (Minh-Nhat Phung)Department of Mathematics,...

2021