Base Models for Parabolic Partial Differential Equations

TL;DR

The paper develops a meta-learning framework for base models of parabolic PDEs, leveraging the Feynman-Kac representation and Girsanov's theorem to reuse Monte Carlo samples when PDE parameters change. It introduces the Neural Girsanov Operator (NGO) to approximate likelihood ratios and learns a continuous task space (p_meta) to rapidly adapt PDE solutions across parameter regimes, including extensions to semi-linear and elliptic PDEs. The approach is validated through maximization tasks in generative modeling and a suite of operator-learning experiments across linear, semi-linear, and canonical PDEs, showing favorable accuracy and speed against DeepONet and Euler-Maruyama baselines. The framework provides a principled, estimator-grounded path to parameter-robust PDE solvers with potential benefits for high-dimensional, multi-task settings in physics-informed modeling and finance, though it notes variance challenges for large drifts. Overall, the work contributes a scalable, theory-backed method for rapidly solving parametric parabolic PDEs via a reusable base model and neural operator framework.

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.

Figures (8)

• Figure 1: Schematic of the proposed procedures for maximizing functions (\ref{['fig:schem_gen']}) and solving PDEs with different parameters (\ref{['fig:schem_pde']}). In both scenarios, we reuse a meta-learned base parameterization across different tasks. (\ref{['fig:schem_gen']}) illustrates sampling target densities using a task-specific diffusion for each density. (\ref{['fig:schem_pde']}) illustrates two solutions to linear parabolic PDEs simulated with the same stochastic process based on the proposed method.
• Figure 2: Simulated solutions of the Fokker-Planck equation of an $1d$ OU-process compared with the analytic solution.
• Figure 3: Numerical results for meta-learning generative models for Gaussian distributions where different means correspond to different tasks.
• Figure 4: Comparison of the normalized errors and inference times of $\mathrm{NGO}$, DeepONet (DON), Girsanov (GIR), and Euler-Maruyama (E-M) on linear and semi-linear parabolic PDEs.
• Figure 5: Ablation study on the number of dimensions of linear parabolic PDEs evaluated at the terminal time $T=0.5$. We show the normalized error (top) and inference time (bottom). The first column shows results on random PDEs with a defined basis; the second column shows results on the Fokker-Planck equation of the OU process; the third column shows results on the Black-Scholes equation.

Theorems & Definitions (2)

• proof
• proof