# Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data

arXiv: https://arxiv.org/abs/1901.06314 · HTML: https://mlsanity.com/paper/1901.06314/

## Key facts

- Cited by 262 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Jan 2019.

## Built on by (most cited first, 30 of 262)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [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)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs](https://mlsanity.com/paper/2106.14103/) (2021)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [SNUG: Self-Supervised Neural Dynamic Garments](https://mlsanity.com/paper/2204.02219/) (2022)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Spline-PINN: Approaching PDEs without Data using Fast, Physics-Informed Hermite-Spline CNNs](https://mlsanity.com/paper/2109.07143/) (2021)
- [Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion](https://mlsanity.com/paper/2109.14860/) (2021)
- [Adaptive Training of Grid-Dependent Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2407.17611/) (2024)
- [A novel meta-learning initialization method for physics-informed neural networks](https://mlsanity.com/paper/2107.10991/) (2021)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Physics-Informed Neural Nets for Control of Dynamical Systems](https://mlsanity.com/paper/2104.02556/) (2021)
- [Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data](https://mlsanity.com/paper/2109.12482/) (2021)
- [Adaptive deep density approximation for Fokker-Planck equations](https://mlsanity.com/paper/2103.11181/) (2021)

## Abstract (arXiv)

Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physics-constrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with data-driven models while obeying the constraints of the problem at hand. This work employs a convolutional encoder-decoder neural network approach as well as a conditional flow-based generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse Kullback-Leibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the Boltzmann-Gibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to out-of-distribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.