Score-based free-form architectures for high-dimensional Fokker-Planck equations

Feng Liu, Faguo Wu, Xiao Zhang

Keywords: Fokker-Planck Equations, Normalization Condition, Score Model, Physical Constraints.

TL;DR: We propose a score PDE loss to decouple the normalization condition, allowing for free-form architectures in solving high-dimensional Fokker-Planck equations.

Abstract: Deep learning methods incorporate PDE residuals as the loss function for solving Fokker-Planck equations, and usually impose the proper normalization condition to avoid a trivial solution. However, soft constraints require careful balancing of multi-objective loss functions, and specific network architectures may limit representation capacity under hard constraints. In this paper, we propose a novel framework: Fokker-Planck neural network (FPNN) that adopts a score PDE loss to decouple the score learning and the density normalization into two stages. Our method allows free-form network architectures to model the unnormalized density and strictly satisfy normalization constraints by post-processing. We demonstrate the effectiveness on various high-dimensional steady-state Fokker-Planck (SFP) equations, achieving superior accuracy and over a 20$\times$ speedup compared to state-of-the-art methods. Without any labeled data, FPNNs achieve the mean absolute percentage error (MAPE) of 11.36%, 13.87% and 12.72% for 4D Ring, 6D Unimodal and 6D Multi-modal problems respectively, requiring only 256, 980, and 980 parameters. Experimental results highlights the potential as a universal fast solver for handling more than 20-dimensional SFP equations, with great gains in efficiency, accuracy, memory and computational resource usage.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

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Submission Number: 11675