Neuro-Spectral Architectures for Causal Physics-Informed Networks

TL;DR

NeuSA introduces Neuro-Spectral Architectures, a PINN framework that projects PDE solutions onto a spectral basis, yielding finite-dimensional coefficients that evolve via a Neural ODE. By using a Fourier basis and an analytical initialization via a linearized operator (Fourier multiplier M), NeuSA overcomes spectral bias and enforces causality through sequential time integration of spectral coefficients, while reconstructing the full solution for physics-informed training. Across 2D wave, 2D Burgers, and 1D sine-Gordon benchmarks, NeuSA achieves markedly faster convergence, improved temporal consistency, and superior predictive accuracy compared to standard PINNs and related baselines, even with substantially fewer training steps. The approach offers a mesh-free, causally consistent alternative to traditional PINNs, with strong potential for complex, inhomogeneous PDEs and time extrapolation, and provides a foundation for extending to more challenging systems and geometries.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs). However, standard MLP-based PINNs often fail to converge when dealing with complex initial value problems, leading to solutions that violate causality and suffer from a spectral bias towards low-frequency components. To address these issues, we introduce NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classical spectral methods, designed to solve linear and nonlinear PDEs with variable coefficients. NeuSA learns a projection of the underlying PDE onto a spectral basis, leading to a finite-dimensional representation of the dynamics which is then integrated with an adapted Neural ODE (NODE). This allows us to overcome spectral bias, by leveraging the high-frequency components enabled by the spectral representation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization scheme based on classical methods. We validate NeuSA on canonical benchmarks for linear and nonlinear wave equations, demonstrating strong performance as compared to other architectures, with faster convergence, improved temporal consistency and superior predictive accuracy. Code and pretrained models are available in https://github.com/arthur-bizzi/neusa.

Figures (10)

• Figure 1: We present NeuSA, a theoretically grounded Physics-informed neural architecture. On the left, we compare various models on a wave propagation problem. The dashed lines represent the discontinuities of a stratified heterogeneous medium. NeuSA achieves the lowest relative error (rMSE) and most accurately preserves sharp wavefronts and reflections. On the right, we show the evolution of the relative L2 error during training. NeuSA converges more rapidly and consistently.
• Figure 2: Neuro-Spectral Architecture. Above: The spectral coefficients for the initial conditions $\hat{\mathbf{u}}(0)$, flowing according to a NODE. Below: The spatial input $\mathbf{x}$ being encoded into the spectral basis functions $\mathbf{b}(\mathbf{x})$. Coefficients and bases are then combined to yield the final result.
• Figure 3: Inference in a Neuro-Spectral model. The initial conditions are decomposed into their spectral coefficients, which are propagated in time via a NODE. The time-iterated coefficients are then reconstructed into the solution at later times.
• Figure 4: Results for the wave equation over the Marmousi benchmark. NeuSA is able to achieve a solution that is much closer than reference methods to the ground truth.
• Figure 5: Results from the sine-Gordon equation. Above: the solution versus the ground truth. Below: the residues between them. NeuSA is both faster and more accurate than the baselines.

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• proof