Neuro-symbolic partial differential equation solver

TL;DR

This paper introduces a scalable neuro-symbolic PDE solver that combines neural networks with numerical discretizations, enabling efficient training of surrogate models while maintaining high accuracy and convergence.

Contribution

It presents a novel neural bootstrapping approach that leverages existing numerical discretizations for scalable, mesh-free PDE solving with neural networks.

Findings

Achieves high resolution and optimal scaling in PDE solutions.

Retains accuracy and convergence properties of traditional numerical methods.

Enables efficient training of neural surrogates for complex systems.

Abstract

We present a highly scalable strategy for developing mesh-free neuro-symbolic partial differential equation solvers from existing numerical discretizations found in scientific computing. This strategy is unique in that it can be used to efficiently train neural network surrogate models for the solution functions and the differential operators, while retaining the accuracy and convergence properties of state-of-the-art numerical solvers. This neural bootstrapping method is based on minimizing residuals of discretized differential systems on a set of random collocation points with respect to the trainable parameters of the neural network, achieving unprecedented resolution and optimal scaling for solving physical and biological systems.