AI paradigm for solving differential equations: first-principles data generation and scale-dilation operator AI solver

TL;DR

This paper introduces a novel AI framework for solving differential equations by generating first-principles data and employing a scale-dilation operator AI solver, effectively addressing high-frequency component approximation issues and enabling broad applicability.

Contribution

It presents a new data generation methodology and a scale-dilation operator AI solver that improve accuracy and efficiency in solving diverse differential equations.

Findings

Achieves superior accuracy over state-of-the-art methods.

Fixes high-frequency component approximation challenges.

Provides a smoother loss landscape for efficient training.

Abstract

Many problems are governed by differential equations (DEs). Artificial intelligence (AI) is a new path for solving DEs. However, data is very scarce and existing AI solvers struggle with approximation of high frequency components (AHFC). We propose an AI paradigm for solving diverse DEs, including DE-ruled first-principles data generation methodology and scale-dilation operator (SDO) AI solver. Using either prior knowledge or random fields, we generate solutions and then substitute them into the DEs to derive the sources and initial/boundary conditions through balancing DEs, thus producing arbitrarily vast amount of, first-principles-consistent training datasets at extremely low computational cost. We introduce a reversible SDO that leverages the Fourier transform of the multiscale solutions to fix AHFC, and design a spatiotemporally coupled, attention-based Transformer AI solver of DEs with SDO. An upper bound on the Hessian condition number of the loss function is proven to be proportional to the squared 2-norm of the solution gradient, revealing that SDO yields a smoother loss landscape, consequently fixing AHFC with efficient training. Extensive tests on diverse DEs demonstrate that our AI paradigm achieves consistently superior accuracy over state-of-the-art methods. This work makes AI solver of DEs to be truly usable in broad nature and engineering fields.