Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods

Chandler Haight; Alex D Rodriguez; Svetlana Roudenko

math.NAOct 20, 2026

Soliton Profiles in 1D: Enhanced Neural-Network Solvers and Constrained Deep Ritz Methods

Authors: Chandler Haight, Alex D Rodriguez, Svetlana Roudenko

Abstract

We develop and compare enhanced neural-network solvers for one-dimensional ground-state profiles, which are solutions of $Q^{\prime\prime}=bQ-γQ^{p},\quad Q>0,\quad p>1, \quad b, γ>0,$ the stationary profile equation for the nonlinear Schrödinger, generalized Korteweg-de Vries, and nonlinear Klein-Gordon equations. Its explicit sech-type solutions serve as exact reference for error computation and benchmark hyperparameter selection. We improve on the standard Adam-trained physics-informed neural network (PINN) baseline by introducing higher-order optimization via Gauss-Newton/Levenberg-Marquardt and L-BFGS refinements, as well as Fourier-feature PINNs, and constrained Deep Ritz (DR) formulations. In DR approaches we introduce two new neural-network formulations, which implement two classical variational characterizations: a Nehari-constrained Deep Ritz method, in which the height is learned by projecting onto the Nehari/Pokhozhaev manifold, and a Weinstein Deep Ritz method, in which a normalized profile minimizes the Weinstein quotient. The Deep Ritz formulations perform best overall, in particular, the Weinstein variant gives the best error-runtime balance. The neural-network based solvers remain less accurate and efficient than classical Petviashvili iteration in 1D, however, the current study provides a proof of concept: the mesh-free formulation, non-reliance on exact profiles or the related features make them promising in higher dimensions and for more complicated nonlinearities and dispersive operators.

Figures & tables

Q_{\theta}(x)=q_{\theta}(x).

Algorithm 1 PINN Method with Second-Order Optimizers for

Q^{\prime\prime}=bQ-\gamma Q^{p}

Parameter	Values tested / description
Domain length $L$	$30$
Pure L–BFGS runs	L–BFGS initialized directly from the network initialization
Adam–L–BFGS runs	Adam warm-up followed by L–BFGS refinement
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$
Number of hidden layers	$3$ , $4$ , or $5$
Neurons per layer	$32$ , $64$ , or $96$

Table 1. Schedule quantities varied in the L–BFGS and Adam–L–BFGS optimizer tests.

Optimizer	Activation	Hidden Layer	Neurons per Layer	$N_{f}$	Adam Warm-up	L–BFGS Steps	History size	Max Iters per Step	L–BFGS LR	$\lambda_{\mathrm{BC}}$
L–BFGS	$\tanh(x)$	$4$	$32$	$2^{10}$	–	$50$	$100$	$10$	$1.0$	$5.0$
Adam–L–BFGS	$\tanh(x)$	$4$	$64$	$2^{9}$	$15{,}000$	$25$	$100$	$5$	$1.0$	$1.0$

Table 2. Selected L–BFGS and Adam–L–BFGS for Dirichlet boundary conditions.

	PINN with L–BFGS optimization				PINN with Adam–L–BFGS optimization
$p$	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	1.314525e-08	15.328056	7.162973e-03	5.864162e-03	5.917457e-08	353.818283	7.896724e-03	6.463914e-03
5	1.679169e-07	14.411380	8.026411e-03	7.032712e-03	2.705050e-07	338.015580	5.668667e-03	4.915231e-03
7	9.080274e-08	10.815290	2.407888e-03	2.187603e-03	2.613929e-07	326.965276	2.374574e-03	2.225127e-03

Table 3. Comparison of PINN performance with L–BFGS and Adam–L–BFGS optimization and Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 1 — Figure 1. Training loss (left) and ground state solution profile convergence (right) for L–BFGS PINN (with parameters as seen in Table 2 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ).

Panel of Figure 2 — Figure 2. Training loss (left) and ground state solution profile convergence (right) for Adam–L–BFGS PINN (with parameters as seen in Table 2 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $15{,}000$ Adam steps followed by $25$ L–BFGS steps.

Parameter	Values tested / description
Domain length $L$	$30$
Pure GN/LM runs	GN/LM initialized directly from the network initialization
Adam–GN/LM runs	Adam warm-up followed by GN/LM refinement
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$
Number of hidden layers	$3$ , $4$ , or $5$
Neurons per layer	$32$ , $64$ , or $96$

Table 4. Schedule and damping quantities varied in the GN/LM and Adam–GN/LM optimizer tests.

Optimizer	Activation	Hidden Layer	Neurons per Layer	$N_{f}$	Adam Warm-up	GN/LM Steps	Max Backtracks	Damping Shrink	Damping Growth	Initial Damping	$\lambda_{\mathrm{BC}}$
GN/LM	$\operatorname{sech}(x)$	$4$	$64$	$2^{9}$	–	$150$	$2$	$0.6$	$10$	$10^{-3}$	$5$
Adam–GN/LM	$\tanh(x)$	$5$	$32$	$2^{9}$	$15{,}000$	$150$	$2$	$0.6$	$10$	$10^{-3}$	$5$

Table 5. Selected GN/LM and Adam–GN/LM configurations for Dirichlet boundary conditions.

	PINN with GN/LM optimization				PINN with Adam–GN/LM optimization
$p$	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	2.763789e-11	4532.584975	1.193728e-04	9.808560e-05	6.763931e-13	1639.024311	1.417711e-04	1.158408e-04
5	1.015142e-10	4940.055658	6.839973e-05	5.982481e-05	1.328934e-10	1872.362646	5.831789e-04	5.065382e-04
7	3.184690e-08	5205.814115	2.681668e-03	2.388390e-03	4.723684e-10	1812.308855	9.558302e-04	8.583382e-04

Table 6. Comparison of PINN performance with GN/LM and Adam–GN/LM optimization and Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 3 — Figure 3. Training loss (left) and ground state solution profile convergence (right) for GN/LM PINN (with parameters as seen in Table 5 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ).

Panel of Figure 4 — Figure 4. Training loss (left) and ground state solution profile convergence (right) for Adam–GN/LM PINN (with parameters as seen in Table 5 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $15{,}000$ Adam steps followed by $150$ GN/LM steps.

	PINN with L–BFGS optimization				PINN with Adam–L–BFGS optimization
$p$	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	5.329101e-08	7.505031	6.583645e-03	5.421850e-03	3.593178e-08	62.361742	3.780430e-03	3.079454e-03
5	7.202707e-08	6.157369	5.845897e-03	5.090068e-03	1.164522e-07	63.238091	2.992726e-03	2.691600e-03
7	4.067500e-08	6.788750	1.609846e-03	1.493517e-03	2.353134e-07	61.959399	9.146505e-04	1.052700e-03

Table 7. Secondary-device GPU+CPU comparison of PINN performance with L–BFGS and Adam–L–BFGS optimization and Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

	PINN with GN/LM optimization				PINN with Adam–GN/LM optimization
$p$	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	4.025347e-11	538.109242	9.699502e-05	7.902533e-05	1.294946e-12	506.161066	5.594483e-05	4.555836e-05
5	4.330355e-11	534.144825	2.525764e-04	2.192186e-04	1.701413e-10	511.624151	5.432745e-04	4.717845e-04
7	5.115845e-06	538.628306	1.256511	1.411610	1.485399e-10	531.554408	6.169997e-04	5.539982e-04

Table 8. Secondary-device GPU+CPU comparison of PINN performance with GN/LM and Adam–GN/LM optimization and Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

	PINN with GN/LM optimization
$p$	$\mathcal{L}_{\mathrm{PINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	2.483710e-08	591.697312	1.413589e+00	1.414096e+00
5	1.058034e-06	586.388381	1.318001e+00	1.417144e+00
7	1.221674e-09	574.305437	6.810766e-04	6.118432e-04

Table 9. Secondary-device GPU+CPU comparison of PINN performance with GN/LM with larger initial damping (

10^{-1}

) and Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 5 — Figure 5. Secondary device ground state solution profile convergence for GN/LM PINN with $p=7$ with parameters as seen in Table 5 (left) and $p=5$ with GN/LM initial damping $10^{-1}$ (right) for $Q^{\prime\prime}=Q-Q^{p}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ).

\xi=\frac{x}{L}.

Algorithm 2 Fourier-feature PINN with Second-Order Optimizer for

Q^{\prime\prime}=bQ-\gamma Q^{p}

Parameter	Values tested / Description
Domain length $L$	$30$
Boundary condition	Dirichlet
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$
Number of hidden layers	$3$ , $4$ , or $5$
Neurons per layer	$32$ , $64$ , or $96$
Optimizer	Adam, L–BFGS, GN/LM, Adam–L–BFGS, or Adam–GN/LM

Table 10. FF–PINN architecture and optimization hyperparameters tested for solving ( 10 ) on

[0,30]

Optimizer	Activation	Hidden Layers	Neurons per Layer	$N_{f}$	Adam Warm-up	$N_{\mathrm{FF}}$	$\sigma_{\mathrm{FF}}$	Train $B$	Include Raw	L–BFGS Steps	History Size	Internal Iteration	L–BFGS LR
Adam–L–BFGS	$\mathrm{SiLU}(x)$	$4$	$32$	$2^{9}$	$2{,}500$	$32$	$0.25$	yes	no	$75$	$20$	$30$	$1.0$

Table 11. Selected FF–PINN configurations for Dirichlet boundary conditions.

	FF–PINN with Dirichlet boundary conditions
$p$	$\mathcal{L}_{\mathrm{FFPINN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	1.756630e-07	219.333194	5.093381e-04	6.852868e-04
5	1.004573e-07	200.046171	4.623447e-04	7.603941e-04
7	5.792016e-08	234.282703	3.989951e-04	7.267693e-04

Table 12. Comparison of FF–PINN performance with Dirichlet 16 boundary conditions (with parameters as in Table 11 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 6 — Figure 6. Training loss (left) and ground state solution profile convergence (right) for FF–PINN (with parameters as seen in Table 11 ) with $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $2{,}500$ Adam steps followed by $75$ L–BFGS steps.

	FF–PINN with Dirichlet boundary conditions
$p$	$\mathcal{L}_{\mathrm{FFPINN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	1.284711e-07	67.670952	7.727327e-04	1.107474e-03
5	1.017865e-07	67.329122	4.419572e-04	7.254083e-04
7	6.116914e-08	72.062521	2.975830e-04	5.681980e-04

Table 13. Secondary-device GPU+CPU comparison of FF–PINN performance with Dirichlet boundary conditions 16 (with parameters as in Table 11 ) for

Q^{\prime\prime}=Q-Q^{p}

\frac{b}{2}Q_{0}^{2}-\frac{\gamma}{p+1}Q_{0}^{p+1}=0,

Algorithm 3 Constrained Deep Ritz Method with Pokhozhaev Penalties

Parameter	Values tested / Description
Domain length $L$	$30$
Boundary condition	Dirichlet
Trial function	Soft penalty trial, hard or decaying Dirichlet ansatz
Optimizer	Adam, L–BFGS, or Adam–L–BFGS
Refinement strategy	GN/LM or Adam–GN/LM
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$

Table 14. Deep Ritz architecture, constraint, and optimization hyperparameters tested for solving ( 10 ) on

[0,30]

Constraint	Optimization	Refinement	Activation	Hidden Layers	Neurons per Layer	$N_{q}$	Adam Warm-up	GN/LM Steps	Max Backtracks	Damping Shrink	Damping Growth	Initial Damping	Residual Points	Quotient Points	$\lambda_{\mathrm{Pokh}}$
Hard	Adam	GN/LM	$\mathrm{SiLU}(x)$	$5$	$32$	$2^{9}$	$2{,}500$	$200$	$8$	$0.7$	$10$	$10^{-3}$	$128$	$128$	$25$

Table 15. Selected Deep Ritz configurations for Dirichlet boundary conditions ( 16 ).

$p$	$\mathcal{L}_{\mathrm{DR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	6.666667e-01	9.620502e-11	264.092212	4.302695e-06	9.264227e-06
5	6.801746e-01	2.375216e-14	252.688689	1.224911e-05	6.049316e-06
7	6.679167e-01	4.063798e-16	246.570857	3.198027e-03	2.417198e-03

Table 16. Performance of the constrained Deep Ritz method with Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 7 — Figure 7. Training objective (left) and ground state solution profile convergence (right) for constrained Deep Ritz method (with parameters as seen in Table 15 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $2{,}500$ Adam steps followed by $200$ GN/LM steps.

$p$	$\mathcal{L}_{\mathrm{DR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	6.666667e-01	7.300457e-11	41.048747	3.697873e-06	7.913867e-06
5	6.801749e-01	1.826449e-12	38.984522	2.828127e-05	1.107741e-05
7	6.677846e-01	1.326228e-09	42.222312	1.397995e-03	1.007490e-03

Table 17. Secondary-device GPU+CPU performance of the constrained Deep Ritz method with Dirichlet boundary conditions ( 16 ) for

Q^{\prime\prime}=Q-Q^{p}

z=\left(\frac{x}{L}\right)^{2},\hskip 16.38895pt\widetilde{Q}_{\theta}(x)=(1-z)\frac{\rho(q_{\theta}(z))}{\rho(q_{\theta}(0))}.

Algorithm 4 Nehari-Constrained Deep Ritz Method for

Q^{\prime\prime}=bQ-\gamma Q^{p}

Parameter	Values tested / Description
Domain length $L$	$30$
Boundary condition	mixed
Trial function	Soft penalty trial, hard or decaying Dirichlet ansatz
Optimizer	Adam, L–BFGS, or Adam–L–BFGS
Refinement strategy	GN/LM or Adam–GN/LM
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$

Table 18. Nehari-constrained Deep Ritz architecture, constraint, and optimization hyperparameters tested for solving ( 10 ) on

[0,30]

Optimization	Refinement	Activation	Hidden Layers	Neurons per Layer	$N_{q}$	Adam Warm-up	GN/LM Steps	Max Backtracks	Damping Shrink	Damping Growth	Initial Damping	Residual Points	Quotient Points	$\lambda_{\mathrm{Pokh}}$
Adam	GN/LM	$\tanh(x)$	$5$	$64$	$2^{9}$	$10{,}000$	$200$	$6$	$0.7$	$15$	$10^{-3}$	$128$	$128$	$0.1$

Table 19. Selected Nehari-constrained Deep Ritz configuration for mixed boundary conditions ( 17 ).

$p$	$\mathcal{L}_{\mathrm{NDR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	6.666667e-01	2.141298e-13	313.008810	1.690701e-07	2.733932e-07
5	6.801748e-01	1.853534e-11	315.139802	5.223740e-06	4.561398e-06
7	6.677478e-01	1.803535e-09	300.718944	2.275145e-04	1.384767e-04

Table 20. Performance of the Nehari-constrained Deep Ritz method with mixed boundary conditions ( 17 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 8 — Figure 8. Training objective (left) and ground state solution profile convergence (right) for the Nehari-constrained Deep Ritz method (with parameters as seen in Table 19 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $10{,}000$ Adam steps followed by $200$ GN/LM steps.

$p$	$\mathcal{L}_{\mathrm{NDR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	6.666667e-01	5.934479e-13	96.631522	3.255530e-07	5.289161e-07
5	6.801748e-01	1.871966e-11	94.922791	5.221092e-06	4.573099e-06
7	6.677480e-01	2.051331e-09	96.848960	2.730569e-04	1.542854e-04

Table 21. Secondary-device GPU+CPU performance of the Nehari-constrained Deep Ritz method with mixed boundary conditions ( 17 ) for

Q^{\prime\prime}=Q-Q^{p}

\rho(s)=\log(1+e^{s}).

Algorithm 5 Weinstein Deep Ritz Method for

Q^{\prime\prime}=bQ-\gamma Q^{p}

Parameter	Values tested / Description
Domain length $L$	$30$
Boundary condition	mixed
Trial function	Soft penalty trial, hard or decaying Dirichlet ansatz
Optimizer	Adam, L–BFGS, or Adam–L–BFGS
Refinement strategy	GN/LM or Adam–GN/LM
Activation functions	$\tanh(x)$ , $\mathrm{SiLU}(x)$ , $\mathrm{SiLUPLUS}(x)$ , $e^{-x^{2}}$ , $e^{-x^{4}}$ , and $\operatorname{sech}(x)$

Table 22. Weinstein Deep Ritz architecture, constraint, and optimization hyperparameters tested for solving ( 10 ) on

[0,30]

Optimization	Refinement	Activation	Hidden Layers	Neurons per Layer	$N_{q}$	Adam Warm-up	GN/LM Steps	GN/LM Max Backtracks	Damping Shrink	Damping Growth	Initial Damping	Residual Points	Quotient Points
Adam	GN/LM	$\mathrm{SiLU}(x)$	$5$	$32$	$2^{9}$	$5{,}000$	$200$	$6$	$0.5$	$15$	$10^{-3}$	$128$	$128$

Table 23. Selected Weinstein Deep Ritz configurations for mixed boundary conditions 17 .

$p$	$\mathcal{L}_{\mathrm{WDR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	CPU time	$L^{\infty}$ error	$L^{2}$ error
3	2.309401e+00	5.781202e-15	148.970724	4.832268e-08	6.092994e-08
5	2.553766e+00	3.953484e-14	150.616751	5.235757e-06	2.322385e-06
7	2.592439e+00	1.687038e-09	145.797130	2.802301e-04	1.451584e-04

Table 24. Performance of the Weinstein Deep Ritz method with mixed boundary conditions ( 17 ) for

Q^{\prime\prime}=Q-Q^{p}

Panel of Figure 9 — Figure 9. Training objective (left) and ground state solution profile convergence (right) for the Weinstein Deep Ritz method (with parameters as seen in Table 23 ) for $Q^{\prime\prime}=Q-Q^{5}$ . The (black dashed) curve labeled Exact in the right panel is the explicit solution given by ( 14 ). The final learned profile corresponds to $5{,}000$ Adam steps followed by $200$ GN/LM steps.

$p$	$\mathcal{L}_{\mathrm{WDR}}(\theta)$	$\mathcal{J}_{\mathrm{GN}}(\theta)$	GPU+CPU time	$L^{\infty}$ error	$L^{2}$ error
3	2.309401e+00	1.934108e-14	40.420787	9.883258e-08	1.186556e-07
5	2.553766e+00	5.054238e-14	44.468209	5.204065e-06	2.321890e-06
7	2.592439e+00	1.258231e-09	44.377295	3.016073e-04	1.542014e-04

Table 25. Secondary-device GPU+CPU performance of the Weinstein Deep Ritz method with mixed boundary conditions ( 17 ) for

Q^{\prime\prime}=Q-Q^{p}

Neural PDE Solvers

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Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen

Mathematical Institute, University of Oxford · Department of Mathematics and Statistics, Boston University

Neural PDE Solvers Optimization Convergence Analysis PDE Solving Physics-Informed ML Partial Differential Equations

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The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning. In these methods, a neural network is trained to approximate the PDE solution by using (stochastic) gradient descent to minimize the PDE residual of the neural network. Due to the non-convexity of the PDE residual objective function, the trained neural network may, in principle, only converge to a local minimizer of the objective function (which would not be a solution of the PDE). Therefore, there is a longstanding question regarding the mathematical foundations of these algorithms, and it is highly valuable to establish that the trained neural network will converge to the PDE solution. In this paper, we consider a class of semilinear PDEs with nonlinearities in the solution and its first derivative. For this class of PDEs, we prove that neural networks trained with gradient descent to minimize the PDE residual objective function will converge to the PDE solution as the network width and training time

\rightarrow \infty

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