# Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder

arXiv: https://arxiv.org/abs/2404.12267 · HTML: https://mlsanity.com/paper/2404.12267/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 79 references that are in the map.
- First posted on arXiv 18 Apr 2024.

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows](https://mlsanity.com/paper/1906.12320/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Multiplicative Normalizing Flows for Variational Bayesian Neural Networks](https://mlsanity.com/paper/1703.01961/) (2017)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [TossingBot: Learning to Throw Arbitrary Objects with Residual Physics](https://mlsanity.com/paper/1903.11239/) (2019)
- [Hamiltonian Generative Networks](https://mlsanity.com/paper/1909.13789/) (2019)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Neural Variational Inference and Learning in Belief Networks](https://mlsanity.com/paper/1402.0030/) (2014)
- [GraphNVP: An Invertible Flow Model for Generating Molecular Graphs](https://mlsanity.com/paper/1905.11600/) (2019)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Integer Discrete Flows and Lossless Compression](https://mlsanity.com/paper/1905.07376/) (2019)
- [Label-Free Supervision of Neural Networks with Physics and Domain Knowledge](https://mlsanity.com/paper/1609.05566/) (2016)
- [Sylvester Normalizing Flows for Variational Inference](https://mlsanity.com/paper/1803.05649/) (2018)
- [Discrete Flows: Invertible Generative Models of Discrete Data](https://mlsanity.com/paper/1905.10347/) (2019)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Improving Variational Auto-Encoders using Householder Flow](https://mlsanity.com/paper/1611.09630/) (2016)

## Built on by (most cited first, 1 of 1)

- [Memory-Conditioned Flow-Matching for Stable Autoregressive PDE Rollouts](https://mlsanity.com/paper/2602.06689/) (2026)

## Abstract (arXiv)

Physics-integrated generative modeling is a class of hybrid or grey-box modeling in which we augment the the data-driven model with the physics knowledge governing the data distribution. The use of physics knowledge allows the generative model to produce output in a controlled way, so that the output, by construction, complies with the physical laws. It imparts improved generalization ability to extrapolate beyond the training distribution as well as improved interpretability because the model is partly grounded in firm domain knowledge. In this work, we aim to improve the fidelity of reconstruction and robustness to noise in the physics integrated generative model. To this end, we use variational-autoencoder as a generative model. To improve the reconstruction results of the decoder, we propose to learn the latent posterior distribution of both the physics as well as the trainable data-driven components using planar normalizng flow. Normalizng flow based posterior distribution harnesses the inherent dynamical structure of the data distribution, hence the learned model gets closer to the true underlying data distribution. To improve the robustness of generative model against noise injected in the model, we propose a modification in the encoder part of the normalizing flow based VAE. We designed the encoder to incorporate scaled dot product attention based contextual information in the noisy latent vector which will mitigate the adverse effect of noise in the latent vector and make the model more robust. We empirically evaluated our models on human locomotion dataset [33] and the results validate the efficacy of our proposed models in terms of improvement in reconstruction quality as well as robustness against noise injected in the model.