Neuro-Spectral Architectures for Causal Physics-Informed Networks

Physics-Informed Neural Networks (PINNs) have emerged as a powerful frame-work for solving partial differential equations (PDEs). However, standard MLP-based PINNs often fail to converge when dealing with complex initial valueproblems, leading to solutions that violate causality and suffer from a spectralbias towards low-frequency components.

To address these issues, we introduce NeuSA (Neuro-Spectral Architectures), a novel class of PINNs inspired by classi-cal spectral methods, designed to solve linear and nonlinear PDEs with variablecoefficients. NeuSA learns a projection of the underlying PDE onto a spectralbasis, leading to a finite-dimensional representation of the dynamics which is thenintegrated with an adapted Neural ODE (NODE).

This allows us to overcome spectralbias, by leveraging the high-frequency components enabled by the spectralrepresentation; to enforce causality, by inheriting the causal structure of NODEs, and to start training near the target solution, by means of an initialization schemebased on classical methods. We validate NeuSA on canonical benchmarks for lin-ear and nonlinear wave equations, demonstrating strong performance as comparedto other architectures, with faster convergence, improved temporal consistencyand superior predictive accuracy.

Code and pretrained models are available inhttps://github.com/arthur-bizzi/neusa.