Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs

TL;DR

Anant-Net is a scalable, interpretable neural framework that effectively solves high-dimensional PDEs, overcoming the curse of dimensionality with high accuracy and efficiency, demonstrated on problems up to 300 dimensions.

Contribution

We introduce Anant-Net, a novel neural surrogate architecture that efficiently handles high-dimensional PDEs and incorporates interpretability through Kolmogorov-Arnold networks.

Findings

Successfully solves 300-dimensional PDEs on a single GPU

Achieves high accuracy and robustness across various equations

Outperforms existing methods in accuracy and runtime

Abstract

High-dimensional partial differential equations (PDEs) arise in diverse scientific and engineering applications but remain computationally intractable due to the curse of dimensionality. Traditional numerical methods struggle with the exponential growth in computational complexity, particularly on hypercubic domains, where the number of required collocation points increases rapidly with dimensionality. Here, we introduce Anant-Net, an efficient neural surrogate that overcomes this challenge, enabling the solution of PDEs in high dimensions. Unlike hyperspheres, where the internal volume diminishes as dimensionality increases, hypercubes retain or expand their volume (for unit or larger length), making high-dimensional computations significantly more demanding. Anant-Net efficiently incorporates high-dimensional boundary conditions and minimizes the PDE residual at high-dimensional collocation points. To enhance interpretability, we integrate Kolmogorov-Arnold networks into the Anant-Net architecture. We benchmark Anant-Net's performance on several linear and nonlinear high-dimensional equations, including the Poisson, Sine-Gordon, and Allen-Cahn equations, demonstrating high accuracy and robustness across randomly sampled test points from high-dimensional space. Importantly, Anant-Net achieves these results with remarkable efficiency, solving 300-dimensional problems on a single GPU within a few hours. We also compare Anant-Net's results for accuracy and runtime with other state-of-the-art methods. Our findings establish Anant-Net as an accurate, interpretable, and scalable framework for efficiently solving high-dimensional PDEs.