PDEfuncta: Spectrally-Aware Neural Representation for PDE Solution Modeling

TL;DR

PDEfuncta introduces a spectrally-aware neural framework that enhances the modeling of complex PDE solutions with high-frequency features, enabling efficient multi-task learning and generalization in scientific computing.

Contribution

The paper proposes Global Fourier Modulation (GFM) and PDEfuncta, novel methods for spectral-aware neural PDE solution modeling that improve accuracy and scalability over prior approaches.

Findings

Enhanced representation of high-frequency PDE features.

Improved generalization to new PDE tasks.

Potential for efficient forward and inverse inference.

Abstract

Scientific machine learning often involves representing complex solution fields that exhibit high-frequency features such as sharp transitions, fine-scale oscillations, and localized structures. While implicit neural representations (INRs) have shown promise for continuous function modeling, capturing such high-frequency behavior remains a challenge-especially when modeling multiple solution fields with a shared network. Prior work addressing spectral bias in INRs has primarily focused on single-instance settings, limiting scalability and generalization. In this work, we propose Global Fourier Modulation (GFM), a novel modulation technique that injects high-frequency information at each layer of the INR through Fourier-based reparameterization. This enables compact and accurate representation of multiple solution fields using low-dimensional latent vectors. Building upon GFM, we introduce PDEfuncta, a meta-learning framework designed to learn multi-modal solution fields and support generalization to new tasks. Through empirical studies on diverse scientific problems, we demonstrate that our method not only improves representational quality but also shows potential for forward and inverse inference tasks without the need for retraining.