# Base Models for Parabolic Partial Differential Equations

arXiv: https://arxiv.org/abs/2407.12234 · HTML: https://mlsanity.com/paper/2407.12234/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 39 references that are in the map.
- First posted on arXiv 17 Jul 2024.

## Builds on (most cited first, 24 of 24)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Fast Sampling of Diffusion Models via Operator Learning](https://mlsanity.com/paper/2211.13449/) (2022)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Meta-learning PINN loss functions](https://mlsanity.com/paper/2107.05544/) (2021)
- [Meta-Auto-Decoder for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2111.08823/) (2021)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2011.04602/) (2020)
- [Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning](https://mlsanity.com/paper/2206.10588/) (2022)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Learning Optimal Control via Forward and Backward Stochastic Differential Equations](https://mlsanity.com/paper/1509.02195/) (2015)

## Built on by (most cited first, 2 of 2)

- [Parabolic Continual Learning](https://mlsanity.com/paper/2503.02117/) (2025)
- [Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guarantees](https://mlsanity.com/paper/2603.19165/) (2026)

## Abstract (arXiv)

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.