# PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations

arXiv: https://arxiv.org/abs/2203.11363 · HTML: https://mlsanity.com/paper/2203.11363/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 40 references that are in the map.
- First posted on arXiv 21 Mar 2022.

## Builds on (most cited first, 19 of 19)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Generative Models and Model Criticism via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/1611.04488/) (2016)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations](https://mlsanity.com/paper/1806.02957/) (2018)
- [PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics](https://mlsanity.com/paper/2106.02993/) (2021)
- [Highly-scalable, physics-informed GANs for learning solutions of stochastic PDEs](https://mlsanity.com/paper/1910.13444/) (2019)
- [Music generation with variational recurrent autoencoder supported by history](https://mlsanity.com/paper/1705.05458/) (2017)
- [A Classifying Variational Autoencoder with Application to Polyphonic Music Generation](https://mlsanity.com/paper/1711.07050/) (2017)

## Built on by (most cited first, 21 of 21)

- [L-HYDRA: Multi-Head Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.02152/) (2023)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Four Principles for Physically Interpretable World Models](https://mlsanity.com/paper/2503.02143/) (2025)
- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing Equations](https://mlsanity.com/paper/2310.04832/) (2023)
- [Geometry-Informed Neural Operator Transformer](https://mlsanity.com/paper/2504.19452/) (2025)
- [InVAErt networks: a data-driven framework for model synthesis and identifiability analysis](https://mlsanity.com/paper/2307.12586/) (2023)
- [PI-VEGAN: Physics Informed Variational Embedding Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/2307.11289/) (2023)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [Physically Interpretable World Models via Weakly Supervised Representation Learning](https://mlsanity.com/paper/2412.12870/) (2024)
- [SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertainty](https://mlsanity.com/paper/2505.13501/) (2025)
- [Adversarial Disentanglement by Backpropagation with Physics-Informed Variational Autoencoder](https://mlsanity.com/paper/2506.13658/) (2025)
- [Variational Rank Reduction Autoencoders for Generative Thermal Design](https://mlsanity.com/paper/2509.08515/) (2025)
- [PartitionVAE -- a human-interpretable VAE](https://mlsanity.com/paper/2302.03689/) (2023)
- [Physics informed Neural Networks applied to the description of wave-particle resonance in kinetic simulations of fusion plasmas](https://mlsanity.com/paper/2308.12312/) (2023)
- [Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equations](https://mlsanity.com/paper/2503.18012/) (2025)
- [Physically-Grounded Goal Imagination: Physics-Informed Variational Autoencoder for Self-Supervised Reinforcement Learning](https://mlsanity.com/paper/2511.06745/) (2025)
- [High-Fidelity Modeling of Stochastic Chemical Dynamics on Complex Manifolds: A Multi-Scale SIREN-PINN Framework for the Curvature-Perturbed Ginzburg-Landau Equation](https://mlsanity.com/paper/2601.08104/) (2026)
- [Target-Aware Linear Regression Under Distribution Shift](https://mlsanity.com/paper/2606.22775/) (2026)
- [Structure-preserving uncertainty quantification for GENERIC dynamics](https://mlsanity.com/paper/2608.12624/) (2026)
- [CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations](https://mlsanity.com/paper/2610.04405/) (2026)

## Abstract (arXiv)

We propose a new class of physics-informed neural networks, called physics-informed Variational Autoencoder (PI-VAE), to solve stochastic differential equations (SDEs) or inverse problems involving SDEs. In these problems the governing equations are known but only a limited number of measurements of system parameters are available. PI-VAE consists of a variational autoencoder (VAE), which generates samples of system variables and parameters. This generative model is integrated with the governing equations. In this integration, the derivatives of VAE outputs are readily calculated using automatic differentiation, and used in the physics-based loss term. In this work, the loss function is chosen to be the Maximum Mean Discrepancy (MMD) for improved performance, and neural network parameters are updated iteratively using the stochastic gradient descent algorithm. We first test the proposed method on approximating stochastic processes. Then we study three types of problems related to SDEs: forward and inverse problems together with mixed problems where system parameters and solutions are simultaneously calculated. The satisfactory accuracy and efficiency of the proposed method are numerically demonstrated in comparison with physics-informed generative adversarial network (PI-WGAN).