DeePoly: A High-Order Accuracy Scientific Machine Learning Framework for Function Approximation and Solving PDEs
This work addresses the need for more reliable and efficient PDE solvers in scientific computing, representing an incremental improvement over existing methods.
The paper tackles the problem of machine learning methods for solving PDEs lacking convergence guarantees and efficiency by introducing DeePoly, a framework that combines DNNs for global features with polynomial bases for local corrections, achieving high-order accuracy and efficiency in numerical experiments.
Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence guarantees and computational efficiency compared to traditional numerical schemes. This work introduces DeePoly, a novel framework that transforms the solution paradigm from pure non-convex parameter optimization to a two-stage approach: first employing a DNN to capture complex global features, followed by linear space optimization with combined DNN-extracted features (Spotter) and polynomial basis functions (Sniper). This strategic combination leverages the complementary strengths of both methods -- DNNs excel at approximating complex global features (i.e., high-gradient features) and stabilize the polynomial approximation while polynomial bases provide high-precision local corrections with convergence guarantees. Theoretical analysis and numerical experiments demonstrate that this approach significantly enhances both high-order accuracy and efficiency across diverse problem types while maintaining mesh-free and scheme-free properties. This paper also serves as a theoretical exposition for the open-source project DeePoly.