# Disentangled Representation Learning for Parametric Partial Differential Equations

arXiv: https://arxiv.org/abs/2410.02136 · HTML: https://mlsanity.com/paper/2410.02136/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 52 of its 81 references that are in the map.
- First posted on arXiv 3 Oct 2024.

## Builds on (most cited first, 30 of 52)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Challenging Common Assumptions in the Unsupervised Learning of Disentangled Representations](https://mlsanity.com/paper/1811.12359/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [NVAE: A Deep Hierarchical Variational Autoencoder](https://mlsanity.com/paper/2007.03898/) (2020)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Variational Autoencoders and Nonlinear ICA: A Unifying Framework](https://mlsanity.com/paper/1907.04809/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Nonlinear ICA Using Auxiliary Variables and Generalized Contrastive Learning](https://mlsanity.com/paper/1805.08651/) (2018)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [A Brief Review of Hypernetworks in Deep Learning](https://mlsanity.com/paper/2306.06955/) (2023)
- [Learning Deep Disentangled Embeddings with the F-Statistic Loss](https://mlsanity.com/paper/1802.05312/) (2018)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Accountability in an Algorithmic Society: Relationality, Responsibility, and Robustness in Machine Learning](https://mlsanity.com/paper/2202.05338/) (2022)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [Unsupervised Model Selection for Variational Disentangled Representation Learning](https://mlsanity.com/paper/1905.12614/) (2019)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)

## Built on by (most cited first, 3 of 3)

- [Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery](https://mlsanity.com/paper/2505.23106/) (2025)
- [A Unified Framework for Simultaneous Parameter and Function Discovery in Differential Equations](https://mlsanity.com/paper/2505.16996/) (2025)
- [SynAct: Synergistic Active Feedback Learning for LLM Agents](https://mlsanity.com/paper/2610.12651/) (2026)

## Abstract (arXiv)

Neural operators (NOs) excel at learning mappings between function spaces, serving as efficient forward solution approximators for PDE-governed systems. However, as black-box solvers, they offer limited insight into the underlying physical mechanism, due to the lack of interpretable representations of the physical parameters that drive the system. To tackle this challenge, we propose a new paradigm for learning disentangled representations from NO parameters, thereby effectively solving an inverse problem. Specifically, we introduce DisentangO, a novel hyper-neural operator architecture designed to unveil and disentangle latent physical factors of variation embedded within the black-box neural operator parameters. At the core of DisentangO is a multi-task NO architecture that distills the varying parameters of the governing PDE through a task-wise adaptive layer, alongside a variational autoencoder that disentangles these variations into identifiable latent factors. By learning these disentangled representations, DisentangO not only enhances physical interpretability but also enables more robust generalization across diverse systems. Empirical evaluations across supervised, semi-supervised, and unsupervised learning contexts show that DisentangO effectively extracts meaningful and interpretable latent features, bridging the gap between predictive performance and physical understanding in neural operator frameworks.