🤖 AI Summary
Solving high-dimensional partial differential equations (PDEs) numerically has long suffered from the “curse of dimensionality.” This paper proposes the Finite Expression Method (FEX), which approximates PDE solutions within a space of finite analytic expressions, using expression complexity—not parameter count—as the fundamental approximation dimension, thereby theoretically circumventing dimensional dependence. FEX integrates deep reinforcement learning–driven symbolic expression search, differentiable symbolic computation, and neural network–assisted fitting to yield explicit, structurally controllable, and physically interpretable approximations. Experiments demonstrate that FEX achieves machine precision across diverse high-dimensional PDEs, with memory complexity scaling only polynomially in dimension—substantially outperforming conventional grid-based methods and neural operators. To our knowledge, FEX is the first approach to simultaneously achieve high-dimensional approximation capability, computational efficiency, and model interpretability.
📝 Abstract
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the"curse of dimensionality"in designing numerical schemes that scale in dimension. This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution.
