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

2025-07-30

AI Summary

Addressing two key bottlenecks in AI-based differential equation solving—data scarcity and the difficulty of approximating high-frequency components (AHFC)—this paper proposes a novel physics-informed AI paradigm. Methodologically, it introduces: (1) a first-principles-driven infinite data generation mechanism that eliminates reliance on ground-truth labeled data; (2) a reversible scale-expansion operator that explicitly enhances model capacity for representing high-frequency solution features; and (3) an end-to-end architecture integrating Fourier multi-scale analysis, spatiotemporal-coupled attention, and Transformer modules. Extensive experiments across diverse partial differential equations demonstrate substantial improvements in convergence speed, generalization capability, and solution accuracy—consistently outperforming state-of-the-art methods. The framework establishes a scalable, robust pathway for high-fidelity, AI-driven scientific modeling in computational science.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Humans and AI: Other Foundations of Human Computation & AI
• Computer Vision: Diffusion Models for Vision

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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.

Problem

• AI paradigm solves differential equations with scarce data
• Proposes first-principles data generation for diverse DEs
• Introduces scale-dilation operator to fix high-frequency approximation

Innovation

• First-principles data generation for DEs
• Scale-dilation operator AI solver
• Fourier-transform-based high-frequency correction