# A physics-informed operator regression framework for extracting data-driven continuum models

arXiv: https://arxiv.org/abs/2009.11992 · HTML: https://mlsanity.com/paper/2009.11992/

## Key facts

- Cited by 60 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2020.

## Built on by (most cited first, 30 of 60)

- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Lie Point Symmetry Data Augmentation for Neural PDE Solvers](https://mlsanity.com/paper/2202.07643/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Blending Neural Operators and Relaxation Methods in PDE Numerical Solvers](https://mlsanity.com/paper/2208.13273/) (2022)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [DoMINO: A Decomposable Multi-scale Iterative Neural Operator for Modeling Large Scale Engineering Simulations](https://mlsanity.com/paper/2501.13350/) (2025)
- [Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs](https://mlsanity.com/paper/2208.01565/) (2022)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [A Neural PDE Solver with Temporal Stencil Modeling](https://mlsanity.com/paper/2302.08105/) (2023)
- [An operator learning perspective on parameter-to-observable maps](https://mlsanity.com/paper/2402.06031/) (2024)
- [Guiding continuous operator learning through Physics-based boundary constraints](https://mlsanity.com/paper/2212.07477/) (2022)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Deep Equilibrium Based Neural Operators for Steady-State PDEs](https://mlsanity.com/paper/2312.00234/) (2023)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)
- [Energy-Dissipative Evolutionary Deep Operator Neural Networks](https://mlsanity.com/paper/2306.06281/) (2023)
- [Finite Element Operator Network for Solving Elliptic-type parametric PDEs](https://mlsanity.com/paper/2308.04690/) (2023)
- [Neural Operator with Regularity Structure for Modeling Dynamics Driven by SPDEs](https://mlsanity.com/paper/2204.06255/) (2022)

## Abstract (arXiv)

The application of deep learning toward discovery of data-driven models requires careful application of inductive biases to obtain a description of physics which is both accurate and robust. We present here a framework for discovering continuum models from high fidelity molecular simulation data. Our approach applies a neural network parameterization of governing physics in modal space, allowing a characterization of differential operators while providing structure which may be used to impose biases related to symmetry, isotropy, and conservation form. We demonstrate the effectiveness of our framework for a variety of physics, including local and nonlocal diffusion processes and single and multiphase flows. For the flow physics we demonstrate this approach leads to a learned operator that generalizes to system characteristics not included in the training sets, such as variable particle sizes, densities, and concentration.