Base Models for Parabolic Partial Differential Equations

TL;DR

This paper introduces a meta-learning framework for efficiently solving parametric parabolic PDEs across different scenarios, leveraging an underlying base distribution to improve generalization and reduce computation time.

Contribution

It proposes a novel meta-learning approach to solve parametric PDEs by learning a base distribution, enabling rapid adaptation to new parameters with extensive experimental validation.

Findings

Improved generalization to new PDE parameter regimes

Reduced computational effort in solving PDEs

Effective application in finance, control, and generative modeling

Abstract

Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often necessary to compute the solutions or a function of the solutions to a parametric PDE in multiple scenarios corresponding to different parameters of this PDE. This process often requires resolving the PDEs from scratch, which is time-consuming. To better employ existing simulations for the PDEs, we propose a framework for finding solutions to parabolic PDEs across different scenarios by meta-learning an underlying base distribution. We build upon this base distribution to propose a method for computing solutions to parametric PDEs under different parameter settings. Finally, we illustrate the application of the proposed methods through extensive experiments in generative modeling, stochastic control, and finance. The empirical results suggest that the proposed approach improves generalization to solving PDEs under new parameter regimes.