NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures

Márcio Marques; Leonardo Mendonça; Leonardo M. Moreira; Christian Júnior de Oliveira; Vitor Balestro; Tiago Novello; Daniel Yukimura; Pavel Petrov

cs.LGSep 25, 2026

NEXT: Physics-Informed Neuro-Spectral Exponential Time Differencing Architectures

Authors: Márcio Marques, Leonardo Mendonça, Leonardo M. Moreira, Christian Júnior de Oliveira, Vitor Balestro, Tiago Novello, Daniel Yukimura, Pavel Petrov,

Organizations: Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brasil · Universidade do Estado do Rio de Janeiro (UERJ), Rio de Janeiro, Brasil · Universidade Federal Fluminense (UFF), Niterói, Brasil

Abstract

Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and lack of causality. Neuro-Spectral Architectures (NeuSA), a recently proposed alternative to PINNs, mitigate both issues, but their numerical integration becomes unstable for stiff differential equations arising in many relevant physical problems. This study proposes Neuro-Spectral Exponential Time Differencing Architectures (NEXT), which combines the spectral representation of the PDE solution in NeuSA with high-order exponential integrators. Within this approach, the linear stiff part of the vector field induced by the PDE is integrated exactly through matrix exponentials, while the possibly nonlinear remainder is modeled by a neural network. The effectiveness of NEXT is verified through benchmark experiments on a set of stiff PDEs, in which NEXT is stable and accurate while NeuSA diverges numerically. It is also shown that NEXT can be applied to inverse problems, where the model has to learn unknown parameters or boundary conditions from sparse data. All code used in this work is publicly available at: https://github.com/marcioh2m/next.git .

Figures & tables

Figure 1: NEXT combines the spectral fidelity and causality of NeuSA with the robustness to stiffness of exponential integrators. This makes it particularly suitable for stiff time-dependent PDEs with sharp interfaces, such as the heterogenous heat equation shown above.

Figure 2: NEXT consists of a recurrent architecture in which the state u is decomposed into basis spectral components $\hat{\textbf{u}}_{n}$ . Each component evolves according to an ODE whose linear part is integrated exactly via the matrix exponential $e^{\Delta tL}$ , while the nonlinear term is handled by the ETD update. The resulting spectral components $\hat{\textbf{u}}_{n+1}$ are then recombined to recover the solution in the spatial domain, yielding $\mathbf{u}_{n+1}$ .

Model	Heat 1d		Burgers 1d		Wave 1d		KdV 1d		Heat 2d
Model	rRMSE	TT	rRMSE	TT	rRMSE	TT	rRMSE	TT	rRMSE	TT
PINN	0.0346	157.94	0.0475	390.11	0.7789	429.63	0.1411	1147.38	0.0810	297.14
QRes	0.0174	211.68	0.0228	544.32	0.7559	578.93	0.0344	1562.83	0.0217	534.00
FLS	0.0485	163.58	0.0420	399.15	0.7312	441.11	0.1496	1176.14	0.0674	304.23
NEXT	0.0030	77.74	0.0011	174.28	0.0031	135.50	0.0035	944.86	0.0010	229.18

Table 1: Quantitative evaluation on forward PDE benchmarks . We report the mean relative root mean squared error (rRMSE) and training time (TT, in seconds) over 10 independent runs with different random seeds. NEXT consistently achieves lower error while requiring less training time than competing methods. The standard deviations of these quantities are also shown in Appendix B .

Figure 3: Error comparison of the predicted solutions for the one-dimensional heat equation with heterogeneous conductivity.

Figure 4: Solutions obtained by PINN and NEXT compared to the reference solution of the wave equation. Our method stays very close to the reference, while PINN fails to learn the reflected wave.

Figure 5: Time snapshots of the NEXT and QRes solutions for the KdV equation in the dispersive shock regime.

Figure 6: Predicted solutions for the Korteweg–De Vries equation, illustrating that our model more accurately captures the nonlinear dispersive shock.

Model	$\mathbf{\lambda_{1}}$	$\mathbf{\lambda_{2}}$	rRMSE	TT
PINN	$0.506\pm 0.201$	$0.108\pm 0.003$	0.099	405
QRes	$0.805\pm 0.103$	$0.104\pm 0.002$	0.040	548
FLS	$0.509\pm 0.203$	$0.108\pm 0.004$	0.099	407
NEXT	$\mathbf{0.955\pm 0.041}$	${\mathbf{0.101}\pm\mathbf{0.001}}$	$\mathbf{0.009}$	$\mathbf{209}$
True Value	1.000	0.100	-	-

Table 2: Burgers parameter identification. Mean and standard deviation of the learned parameters

\lambda_{1},\,\lambda_{2}

over 10 independent experiments, as well as the mean values of the rRMSE for the solution

u

and the training time.

Figure 7: Predicted heat flux on the inverse heat problem using NEXT.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

	NEXT				Baselines
	LR	N. Steps	Width	Depth	LR	N. Steps
Heat 1D	0.02	500	796	2	0.001	5000
Burgers	0.02	500	800	2	0.001	5000
Wave	0.02	1000	300	2	0.001	5000
Korteweg–De Vries	0.01	1000	400	2	0.001	10000
Heat 2D	0.01	500	100	2	0.001	20000

Appendix

Table 3: Learning rate, number of training steps and architecture shape used in each of the forward experiments.

Model	Heat 1d		Burgers 1d		Wave 1d		KdV 1d		Heat 2d
Model	rRMSE	TT	rRMSE	TT	rRMSE	TT	rRMSE	TT	rRMSE	TT
PINN	0.0049	0.02	0.0187	0.55	0.0682	3.48	0.0552	0.32	0.0125	0.24
QRes	0.0050	0.04	0.0098	1.22	0.0137	6.67	0.0210	0.40	0.0074	0.61
FLS	0.0457	0.04	0.0249	0.64	0.0463	1.57	0.0572	0.33	0.0139	0.59
NEXT	0.0008	0.23	0.0013	0.97	0.0004	1.09	0.0004	2.24	0.0009	0.07

Appendix

Table 4: Quantitative evaluation on PDE benchmarks . Below we list the standard deviation for the relative root mean square error (rRMSE) and training time (TT, in seconds) for all forward problems, across 10 different runs with different random seeds.

Figure 8: The conductivity profile used in the experiment with one-dimensional heat equation ( 9 ).

Figure 9: Wave propagation velocity $c(x)$ for the wave equation’s simulation domain.

Figure 10: The conductivity profile considered in the 2d heat equation.

Figure 11: Predicted solutions for the 2d heat equation. The rRMSE is computed for each time snapshot. NEXT correctly predicts the solution, while the baseline models have difficulty dealing with the sharp interfaces between materials.

Figure 12: Absolute error of each model prediction for the two-dimensional heat equation. The displayed error is the mean absolute error for each time snapshot. The baseline models fail to learn the initial condition, and this error propagates in time, leading to a very inaccurate solution at later times. NEXT, on the other hand, has a more stable error profile, due to its causal nature and to its capacity to automatic enforcement of initial conditions.

Parameters	Unit	Value
$\rho$	kg/m 3	3120
$c_{p}$	J/(kg·K)	850
$k_{0}$	W/(m·K)	4.86
$k_{1}$	W/(m·K 2 )	0.0016
$h$	W/(m 2 ·K)	7
$\sigma$	W/(m 2 ·K 4 )	5.67e-08

Appendix

Table 5: Thermophysical parameters for eq. ( 15 )

Solver	Wave 1D		Heat 2D
Solver	rRMSE	TT	rRMSE	TT
ETD1	0.0629	38.6	0.0031	66.2
ETD4	0.0030	130.0	0.0007	215.7

Appendix

Table 6: Relative root mean square error and training time (in seconds) obtained by NEXT using different orders of exponential integrator.

W \ D	1	2	3	1	2	3
	rRMSE			TT
100	0.0015	0.0008	0.0019	161	236	313
200	0.0015	0.0006	0.0013	214	344	473
300	0.0031	0.0011	-	264	449	-
400	0.0037	0.0025	0.0029	333	804	1274

Appendix

Table 7: NEXT’s performance in the 2D Heat equation by Width (W) and Depth (D) of the neural vector field. A dash (-) indicates training did not converge. The architecture chosen for the experiment reported in the paper (underlined below) was the one that achieved the best trade-off between accuracy and training time.

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Neuro-spectral architectures (NeuSA) are a class of physics-informed neural networks (PINNs) that have proven effective for solving time-dependent PDEs in the frequency domain. However, training and inference often rely on costly numerical integration of the learned vector field, typically using sequential Runge-Kutta time stepping, which limits temporal parallelism. We introduce PaPi, an integration scheme inspired by weakly nonlinear flows that enables time-Parallel inference through exponential integration and a Duhamel-Picard scheme. We show that the Duhamel-Picard iteration converges exponentially for all flows, and provide explicit error bounds that depend on how close the flow is to linear. Experiments show that PaPi accelerates training and inference by up to two orders of magnitude, achieving training times of under one second for low-dimensional PDE learning problems, while maintaining competitive predictive accuracy.

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