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}

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