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arxiv: 2506.04742 · v3 · submitted 2025-06-05 · 🧮 math.OC · cs.AI

Employing Deep Neural Operators for PDE control by decoupling training and optimization

Oliver G. S. Lundqvist , Fabricio Oliveira This is my paper

Pith reviewed 2026-05-19 11:38 UTC · model grok-4.3

classification 🧮 math.OC cs.AI
keywords neural operatorsPDE-constrained controlDeepONetphysics-informed trainingoptimal controlautomatic differentiationdecoupled optimization
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The pith

A neural operator trained once on PDE residuals solves multiple tracking control problems via later optimization.

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

The paper establishes that a neural operator such as DeepONet can be trained in one physics-informed phase that includes a residual penalty term, after which an unconstrained optimizer uses automatic differentiation through the fixed operator to find controls for different tracking targets. This separation removes the need to combine data, residuals, and control costs inside a single training loop. If the approach works as described, it simplifies code for PDE-constrained problems and yields faster iterations on nonlinear time-dependent equations while still satisfying the governing PDE through the penalty.

Core claim

By adding the differential equation residual as a penalty inside the objective, a once-trained neural operator supplies both the state solution and its gradients with respect to the control; an iterative unconstrained optimizer then minimizes the tracking cost while the penalty keeps the PDE approximately satisfied. The same operator applies to new targets without retraining. Benchmarks on Poisson, viscous Burgers, and Stokes problems show that the method reaches competitive accuracy with reduced per-iteration cost for the nonlinear Burgers case and remains feasible for the flow-control example.

What carries the argument

The trained neural operator acting as a fixed, differentiable map from control to state, with the residual penalty inside the optimization objective enforcing the PDE constraint.

If this is right

  • For the viscous Burgers problem the method reaches accuracy comparable to adjoint solvers while cutting iteration time by up to a factor of four.
  • The identical trained operator can be reused on new tracking targets without any retraining step.
  • For the linear Poisson problem the classical adjoint method retains higher final accuracy, indicating the neural route is more advantageous when the PDE is nonlinear or time-dependent.
  • Feasibility for Stokes flow control is confirmed by comparing the optimized control against a reference forward solver.

Where Pith is reading between the lines

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

  • Because the operator is fixed after training, the same surrogate could support repeated optimizations under changing objectives or modest parameter variations with little extra cost.
  • The speed gain observed on time-dependent nonlinear problems suggests the approach may scale to real-time or receding-horizon control settings where many solves are required.
  • If operator accuracy improves with modest increases in network size or training data, the method could extend to higher-dimensional domains where adjoint derivations become cumbersome.

Load-bearing premise

The operator must return solution and gradient values accurate enough that the subsequent unconstrained optimization converges to a control whose PDE residual stays inside the chosen tolerance.

What would settle it

Apply a high-fidelity numerical PDE solver to the final control produced by the method and check whether the residual norm exceeds the tolerance used during training; a large excess would show the constraint is not met.

Figures

Figures reproduced from arXiv: 2506.04742 by Fabricio Oliveira, Oliver G. S. Lundqvist.

Figure 1
Figure 1. Figure 1: The schematic of our proposed method. each defining a different optimal control problem. Similarly, operating a power plant involves distinct objectives during startup, steady-state operation, and shutdown, resulting in various formulations of the cost function depending on the operational context. To address this limitation, we propose a method that fully decouples neural network training from the optimal… view at source ↗
Figure 2
Figure 2. Figure 2: Results for the linear ODE with different cost functions. The top row shows the obtained [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Results for the nonlinear ODE with different cost functions. The top row shows the obtained [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Results for diffusion-reaction with different cost functions. The top row shows the obtained [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Training loss for the DeepONet models. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Random test results for the linear ode model for the GRF function with different length [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Random test results for the nonlinear ode model for the GRF function with different length [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Random test results for the diffusion model for the GRF function with different length [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Sensitvity analysis for different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p028_17.png] view at source ↗
read the original abstract

Neural networks have been applied to control problems, typically by combining data, differential equation residuals, and objective costs in the training loss or by incorporating auxiliary architectural components. Instead, we propose a streamlined approach that decouples the control problem from the training process, rendering these additional layers of complexity unnecessary. In particular, our analysis and computational experiments demonstrate that a simple neural operator architecture, such as DeepONet, coupled with an unconstrained optimization routine, can solve tracking-type partial differential equation (PDE) constrained control problems with a single physics-informed training phase and a subsequent optimization phase. We achieve this by adding a penalty term to the cost function based on the differential equation residual to penalize deviations from the PDE constraint. This allows gradient computations with respect to the control using automatic differentiation through the trained neural operator within an iterative optimization routine, while satisfying the PDE constraints. Once trained, the same neural operator can be reused across different tracking targets without retraining. We benchmark our method on scalar elliptic (Poisson's equation), nonlinear transport (viscous Burgers' equation), and flow (Stokes equation) control problems. For the Poisson and Burgers problems, we compare against adjoint-based solvers: for the time-dependent Burgers problem, the approach achieves competitive accuracy with iteration times up to four times faster, while for the linear Poisson problem, the adjoint method retains superior accuracy, suggesting the approach is best suited to nonlinear and time-dependent settings. For the flow control problem, we verify the feasibility of the optimized control through a reference forward solver.

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

2 major / 2 minor

Summary. The paper claims that tracking-type PDE-constrained control problems can be solved by training a DeepONet once with a physics-informed loss that includes a PDE residual penalty term, then freezing the operator and solving an unconstrained optimization problem whose objective is the tracking cost plus a weighted residual penalty, with gradients with respect to the control obtained by automatic differentiation through the fixed operator. The same trained operator is reused for different target states. Numerical results are presented for Poisson, viscous Burgers, and Stokes control problems, with comparisons to adjoint-based solvers indicating competitive accuracy and up to 4x faster iteration times for the nonlinear time-dependent Burgers case, while the adjoint method is more accurate for linear Poisson; a feasibility check is performed for Stokes flow.

Significance. If the central claim holds, the decoupling of a single physics-informed training phase from subsequent optimization phases offers a practical simplification for PDE control, particularly when the same operator can be reused across multiple tracking targets without retraining. The approach leverages standard DeepONet architectures and off-the-shelf optimizers, and the reported speed advantage on the nonlinear Burgers benchmark suggests potential utility in time-dependent or nonlinear settings where adjoint methods are costly to derive or implement.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (Burgers benchmark): the claim of 'competitive accuracy' with iteration times up to four times faster is not supported by any reported quantitative error tables, relative L2 norms for state or control, or residual values at convergence; without these metrics it is impossible to verify whether the penalty term drives the PDE residual below the tolerance achieved by the adjoint solver.
  2. [§3.2] §3.2 (optimization phase, gradient computation): the method obtains control gradients by automatic differentiation through the learned operator G in the term λ·residual(G(u),u); for the nonlinear viscous Burgers equation this relies on the unproven assumption that local approximation errors in G and its derivatives remain small enough that the optimizer can still enforce the PDE constraint, yet no error bounds, derivative accuracy tests, or out-of-distribution checks on the control measure are provided.
minor comments (2)
  1. [§3] The description of the unconstrained optimizer (e.g., L-BFGS, gradient descent with specific step-size rule) is not stated explicitly in the methods section.
  2. [§4.3] For the Stokes feasibility verification, the manuscript should clarify how the reference forward solver is used to confirm that the optimized control produces a state satisfying the PDE to machine precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments point by point below, indicating where revisions will be made to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (Burgers benchmark): the claim of 'competitive accuracy' with iteration times up to four times faster is not supported by any reported quantitative error tables, relative L2 norms for state or control, or residual values at convergence; without these metrics it is impossible to verify whether the penalty term drives the PDE residual below the tolerance achieved by the adjoint solver.

    Authors: We agree with the referee that quantitative metrics are essential to support the claims of competitive accuracy. In the revised manuscript, we will add a table in §4 presenting the relative L² errors for both the state and control variables, as well as the final PDE residual norms for our method and the adjoint-based solver. This will provide a clear basis for comparing accuracy and verifying the effectiveness of the penalty term. revision: yes

  2. Referee: [§3.2] §3.2 (optimization phase, gradient computation): the method obtains control gradients by automatic differentiation through the learned operator G in the term λ·residual(G(u),u); for the nonlinear viscous Burgers equation this relies on the unproven assumption that local approximation errors in G and its derivatives remain small enough that the optimizer can still enforce the PDE constraint, yet no error bounds, derivative accuracy tests, or out-of-distribution checks on the control measure are provided.

    Authors: We acknowledge that the manuscript does not provide theoretical error bounds or explicit derivative accuracy tests. While deriving rigorous bounds is beyond the scope of the current work, we will include additional numerical validation in the revision. Specifically, we will report results from comparing the automatic differentiation gradients with finite-difference approximations for several control inputs, and perform checks on controls outside the training distribution to assess the robustness of the approach. These additions will help substantiate the practical reliability of the method for the nonlinear case. revision: partial

Circularity Check

0 steps flagged

No significant circularity in decoupling of neural operator training from control optimization

full rationale

The paper trains a DeepONet once using a physics-informed loss that penalizes PDE residuals to approximate the solution operator, then freezes the operator and performs unconstrained minimization of a tracking objective augmented by a residual penalty term, with gradients obtained via automatic differentiation through the fixed operator. This construction does not reduce any claimed performance or accuracy metric to a quantity defined by the same fitted parameters; the training phase produces an independent approximation whose quality is externally validated through benchmarks against adjoint solvers on Poisson, Burgers, and Stokes problems. No self-citation chains, self-definitional equations, or fitted inputs renamed as predictions appear in the derivation. The approach is self-contained against the standard neural-operator and PINN paradigms.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that a single trained neural operator can serve as a reliable differentiable surrogate for the PDE solution map during control optimization; the penalty weight on the residual is an implicit hyperparameter whose value must be chosen to balance accuracy and constraint satisfaction.

free parameters (1)
  • PDE residual penalty weight
    The coefficient multiplying the differential-equation residual term in the training loss is a tunable hyperparameter whose specific value is not stated in the abstract.
axioms (1)
  • domain assumption A neural operator trained on a finite set of input-control pairs can produce sufficiently accurate solution and gradient fields for use inside a gradient-based optimizer.
    This assumption is required for the subsequent unconstrained optimization phase to locate controls that satisfy the PDE to acceptable tolerance.

pith-pipeline@v0.9.0 · 5806 in / 1371 out tokens · 78634 ms · 2026-05-19T11:38:53.559508+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Learning to Control PDEs with Differentiable Predictive Control and Time-Integrated Neural Operators

    cs.CE 2025-11 unverdicted novelty 5.0

    A framework using time-integrated DeepONets inside differentiable predictive control learns fast neural policies that track targets and satisfy constraints on PDEs like heat, Burgers', and reaction-diffusion equations.

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Works this paper leans on

43 extracted references · 43 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    G. S. Christensen, M. E. El-Hawary, and S. A. Soliman. Optimal Control Applications in Electric Power Systems, volume 35 of Mathematical Concepts and Methods in Science and Engineering. Springer, 1987

    work page 1987

  2. [2]

    Optimal control of a distribution system with a virtual power plant

    Roberto Caldon, Andrea Rossi Patria, and Roberto Turri. Optimal control of a distribution system with a virtual power plant. Bulk power system dynamics and control, Cortina. d’Ampezzo, Italy, page 18, 2004

    work page 2004

  3. [3]

    Optimal control of wind power plants

    Maarten Steinbuch, WW De Boer, Okko H Bosgra, SAWM Peters, and Jeroen Ploeg. Optimal control of wind power plants. Journal of Wind Engineering and Industrial Aerodynamics, 27(1):237–246, 1988

    work page 1988

  4. [4]

    Aircraft trajectory optimization and contrails avoidance in the presence of winds

    Banavar Sridhar, Hok K Ng, and Neil Y Chen. Aircraft trajectory optimization and contrails avoidance in the presence of winds. Journal of Guidance, Control, and Dynamics, 34(5):1577–1584, 2011

    work page 2011

  5. [5]

    An optimal-control-based framework for trajectory planning, threat assessment, and semi-autonomous control of passenger vehicles in hazard avoidance scenarios

    Sterling J Anderson, Steven C Peters, Tom E Pilutti, and Karl Iagnemma. An optimal-control-based framework for trajectory planning, threat assessment, and semi-autonomous control of passenger vehicles in hazard avoidance scenarios. International Journal of Vehicle Autonomous Systems, 8(2-4):190–216, 2010

    work page 2010

  6. [6]

    Optimal control with aerospace applications

    James M Longuski, José J Guzmán, and John E Prussing. Optimal control with aerospace applications. Springer, 2014

    work page 2014

  7. [7]

    Optimal control applied to biological models

    Suzanne Lenhart and John T Workman. Optimal control applied to biological models . Chapman and Hall/CRC, 2007

    work page 2007

  8. [8]

    Optimal control by deep learning techniques and its applications on epidemic models

    Shuangshuang Yin, Jianhong Wu, and Pengfei Song. Optimal control by deep learning techniques and its applications on epidemic models. Journal of Mathematical Biology, 86(36), 2023

    work page 2023

  9. [9]

    Optimal control theory and static optimization in economics

    Daniel Leonard and Ngo Van Long. Optimal control theory and static optimization in economics. Cam- bridge University Press, 1992

    work page 1992

  10. [10]

    Stochastic control for economic models: past, present and the paths ahead

    David A Kendrick. Stochastic control for economic models: past, present and the paths ahead. Journal of economic dynamics and control, 29(1-2):3–30, 2005

    work page 2005

  11. [11]

    Optimal control of pdes using physics-informed neural networks

    Saviz Mowlavi and Saleh Nabi. Optimal control of pdes using physics-informed neural networks. Journal of Computational Physics, 473:111731, 2023

    work page 2023

  12. [12]

    Neural network approaches for parameterized optimal control

    Deepanshu Verma, Nick Winovich, Lars Ruthotto, and Bart van Bloemen Waanders. Neural network approaches for parameterized optimal control. arXiv preprint arXiv:2402.10033, 2024

    work page arXiv 2024

  13. [13]

    A pinn approach for the online identification and control of unknown pdes

    Alessandro Alla, Giulia Bertaglia, and Elisa Calzola. A pinn approach for the online identification and control of unknown pdes. arXiv preprint arXiv:2408.03456, 2024

    work page arXiv 2024

  14. [14]

    Real-time optimal control of high- dimensional parametrized systems by deep learning-based reduced order models

    Matteo Tomasetto, Andrea Manzoni, and Francesco Braghin. Real-time optimal control of high- dimensional parametrized systems by deep learning-based reduced order models. arXiv preprint arXiv:2409.05709, 2024

    work page arXiv 2024

  15. [15]

    The admm-pinns algorithmic framework for nonsmooth pde-constrained optimization: A deep learning approach

    Yongcun Song, Xiaoming Yuan, and Hangrui Yue. The admm-pinns algorithmic framework for nonsmooth pde-constrained optimization: A deep learning approach. arXiv preprint arXiv:2302.08309, 2023

    work page arXiv 2023

  16. [16]

    Kingma and J

    D.P. Kingma and J. Ba. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR), 2015

    work page 2015

  17. [17]

    C.G. Broyden. The convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications, 6(3):222–231, 1970

    work page 1970

  18. [18]

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

    Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561, 2017

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  19. [19]

    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 Computational Physics, 378:686–707, 2019

    work page 2019

  20. [20]

    Physics-informed neural networks for inverse problems in supersonic flows

    Ameya D Jagtap, Zhiping Mao, Nikolaus Adams, and George E Karniadakis. Physics-informed neural networks for inverse problems in supersonic flows. Journal of Computational Physics, 448:111402, 2022

    work page 2022

  21. [21]

    Kashefi and T

    A. Kashefi and T. Mukerji. Physics-informed pointnet: A deep learning solver for steady-state incompress- ible flows and thermal fields on multiple sets of irregular geometries. Journal of Computational Physics, 452:110904, 2022. 11

    work page 2022

  22. [22]

    L. Yang, X. Meng, and G. E. Karniadakis. B-pinns: Bayesian physics-informed neural networks for forward and inverse pde problems with noisy data. Journal of Computational Physics, 425:109913, 2021

    work page 2021

  23. [23]

    Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francesc Verdugo, and Steven G. Johnson. Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing, 43(6):B1105–B1132, 2021

    work page 2021

  24. [24]

    An extended physics informed neural network for preliminary analysis of parametric optimal control problems

    Nicola Demo, Maria Strazzullo, and Gianluigi Rozza. An extended physics informed neural network for preliminary analysis of parametric optimal control problems. Computers & Mathematics with Applications, 143:383–396, 2023

    work page 2023

  25. [25]

    L. S. Pontryagin, V . G. Boltyanskii, R. V . Gamkrelidze, and E. F. Mishchenko. On the mathematical theory of optimal processes. Doklady Akademii Nauk SSSR, 111(5):777–778, 1956

    work page 1956

  26. [26]

    Aonn: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems

    Pengfei Yin, Guangqiang Xiao, Kejun Tang, and Chao Yang. Aonn: An adjoint-oriented neural network method for all-at-once solutions of parametric optimal control problems. SIAM Journal on Scientific Computing, 46(1):C127–C153, 2024

    work page 2024

  27. [27]

    Pontryagin neural networks for the class of optimal control problems with integral quadratic cost

    Enrico Schiassi, Francesco Calabrò, and Davide Elia De Falco. Pontryagin neural networks for the class of optimal control problems with integral quadratic cost. Aerospace Research Communications, 2:13151, 2024

    work page 2024

  28. [28]

    Deep multi-input and multi-output operator networks method for optimal control of pdes

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

    work page 2024

  29. [29]

    Deep mixed residual method for solving PDE-constrained optimization problems

    Jinjun Yong, Xianbing Luo, Shuyu Sun, and Changlun Ye. Deep mixed residual method for solving PDE-constrained optimization problems. Computers & Mathematics with Applications , 176:510–524, 2024

    work page 2024

  30. [30]

    Neural operators for pde backstepping control of first-order hyperbolic pide with recycle and delay

    Jie Qi, Jing Zhang, and Miroslav Krstic. Neural operators for pde backstepping control of first-order hyperbolic pide with recycle and delay. Systems & Control Letters, 185:105714, 2024

    work page 2024

  31. [31]

    Solving pde-constrained control problems using operator learning

    Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, and Hyung Ju Hwang. Solving pde-constrained control problems using operator learning. In Proceedings of the Thirty-Sixth AAAI Conference on Artificial Intelligence (AAAI-22), pages 4504–4512, 2022

    work page 2022

  32. [32]

    Learning nonlinear operators via deeponet based on the universal approximation theorem of operators

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

    work page 2021

  33. [33]

    Chen and H

    T. Chen and H. Chen. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Transactions on Neural Networks, 6(4):911–917, 1995

    work page 1995

  34. [34]

    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

    work page 2021

  35. [35]

    Learning the solution operator of parametric partial differential equations with physics-informed deeponets

    Sifan Wang, Hanwen Wang, and Paris Perdikaris. Learning the solution operator of parametric partial differential equations with physics-informed deeponets. Science Advances, 7(40):eabi8605, 2021

    work page 2021

  36. [36]

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

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

    work page 2021

  37. [37]

    A multiple shooting algorithm for direct solution of optimal control problems

    Hans Georg Bock and Karl-Josef Plitt. A multiple shooting algorithm for direct solution of optimal control problems. IFAC Proceedings Volumes, 17(2):1603–1608, 1984

    work page 1984

  38. [38]

    A survey of numerical methods for optimal control

    Anil V Rao. A survey of numerical methods for optimal control. Advances in the astronautical Sciences, 135(1):497–528, 2009

    work page 2009

  39. [39]

    Convergence of the forward-backward sweep method in optimal control

    Michael McAsey, Libin Mou, and Weimin Han. Convergence of the forward-backward sweep method in optimal control. Computational Optimization and Applications, 53(1):207–226, 2012

    work page 2012

  40. [40]

    A computational method on derivative variations of optimal control

    Enkhbat Rentsen, Masaru Kamada, Amr Radwan, and Wejdan Alrashdan. A computational method on derivative variations of optimal control. Journal of Mathematics and Computer Science, 28:203–212, 2023

    work page 2023

  41. [41]

    Hager, Anil V

    Divya Garg, Michael Patterson, William W. Hager, Anil V . Rao, David A. Benson, and Geoffrey T. Hunt- ington. A unified framework for the numerical solution of optimal control problems using pseudospectral methods. Automatica, 46(11):1843–1851, 2010. 12

    work page 2010

  42. [42]

    Optimal control of systems governed by partial differential equations, volume 170

    Jacques Louis Lions. Optimal control of systems governed by partial differential equations, volume 170. Springer, 1971

    work page 1971

  43. [43]

    improved fully connected network

    Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. Pytorch: An imperative style, high-perfo...

    work page 2019