# MetaPhysiCa: OOD Robustness in Physics-informed Machine Learning

arXiv: https://arxiv.org/abs/2303.03181 · HTML: https://mlsanity.com/paper/2303.03181/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 59 references that are in the map.
- First posted on arXiv 6 Mar 2023.

## Builds on (most cited first, 30 of 42)

- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Arbitrary Style Transfer in Real-time with Adaptive Instance Normalization](https://mlsanity.com/paper/1703.06868/) (2017)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Invariant Risk Minimization](https://mlsanity.com/paper/1907.02893/) (2019)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [DAGs with NO TEARS: Continuous Optimization for Structure Learning](https://mlsanity.com/paper/1803.01422/) (2018)
- [Out-of-Distribution Generalization via Risk Extrapolation (REx)](https://mlsanity.com/paper/2003.00688/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Bilevel Programming for Hyperparameter Optimization and Meta-Learning](https://mlsanity.com/paper/1806.04910/) (2018)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks](https://mlsanity.com/paper/2009.11848/) (2020)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)

## Built on by (most cited first, 6 of 6)

- [Out-of-Domain Generalization in Dynamical Systems Reconstruction](https://mlsanity.com/paper/2402.18377/) (2024)
- [Learning Interpretable Hierarchical Dynamical Systems Models from Time Series Data](https://mlsanity.com/paper/2410.04814/) (2024)
- [SineNet: Learning Temporal Dynamics in Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2403.19507/) (2024)
- [ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inference](https://mlsanity.com/paper/2403.10766/) (2024)
- [Learning where to learn: Training data distribution optimization for scientific machine learning](https://mlsanity.com/paper/2505.21626/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)

## Abstract (arXiv)

A fundamental challenge in physics-informed machine learning (PIML) is the design of robust PIML methods for out-of-distribution (OOD) forecasting tasks. These OOD tasks require learning-to-learn from observations of the same (ODE) dynamical system with different unknown ODE parameters, and demand accurate forecasts even under out-of-support initial conditions and out-of-support ODE parameters. In this work we propose a solution for such tasks, which we define as a meta-learning procedure for causal structure discovery (including invariant risk minimization). Using three different OOD tasks, we empirically observe that the proposed approach significantly outperforms existing state-of-the-art PIML and deep learning methods.