# E-PINNs: Epistemic Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2503.19333 · HTML: https://mlsanity.com/paper/2503.19333/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 64 references that are in the map.
- First posted on arXiv 25 Mar 2025.

## Builds on (most cited first, 30 of 31)

- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep Ensembles: A Loss Landscape Perspective](https://mlsanity.com/paper/1912.02757/) (2019)
- [Uncertainty Sets for Image Classifiers using Conformal Prediction](https://mlsanity.com/paper/2009.14193/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Concrete Dropout](https://mlsanity.com/paper/1705.07832/) (2017)
- [Pitfalls of In-Domain Uncertainty Estimation and Ensembling in Deep Learning](https://mlsanity.com/paper/2002.06470/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)
- [High-Quality Prediction Intervals for Deep Learning: A Distribution-Free, Ensembled Approach](https://mlsanity.com/paper/1802.07167/) (2018)
- [Dropout Q-Functions for Doubly Efficient Reinforcement Learning](https://mlsanity.com/paper/2110.02034/) (2021)
- [Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks](https://mlsanity.com/paper/1905.04817/) (2019)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Quality of Uncertainty Quantification for Bayesian Neural Network Inference](https://mlsanity.com/paper/1906.09686/) (2019)
- [Multi-fidelity Bayesian Neural Networks: Algorithms and Applications](https://mlsanity.com/paper/2012.13294/) (2020)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Dropout Inference in Bayesian Neural Networks with Alpha-divergences](https://mlsanity.com/paper/1703.02914/) (2017)
- [Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks](https://mlsanity.com/paper/2402.15406/) (2024)
- [A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems](https://mlsanity.com/paper/2202.04648/) (2022)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification](https://mlsanity.com/paper/2402.15115/) (2024)
- [Artificial intelligence for partial differential equations in computational mechanics: A review](https://mlsanity.com/paper/2410.19843/) (2024)
- [Conformalized Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.08111/) (2024)
- [Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversion](https://mlsanity.com/paper/2303.07392/) (2023)

## Built on by (most cited first, 4 of 4)

- [SPIEDiff: robust learning of long-time macroscopic dynamics from short-time particle simulations with quantified epistemic uncertainty](https://mlsanity.com/paper/2505.13501/) (2025)
- [Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss](https://mlsanity.com/paper/2510.15664/) (2025)
- [Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics](https://mlsanity.com/paper/2601.03673/) (2026)
- [Structure-preserving uncertainty quantification for GENERIC dynamics](https://mlsanity.com/paper/2608.12624/) (2026)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) have demonstrated promise as a framework for solving forward and inverse problems involving partial differential equations. Despite recent progress in the field, it remains challenging to quantify uncertainty in these networks. While techniques such as Bayesian PINNs (B-PINNs) provide a principled approach to capturing epistemic uncertainty through Bayesian inference, they can be computationally expensive for large-scale applications. In this work, we propose Epistemic Physics-Informed Neural Networks (E-PINNs), a framework that uses a small network, the epinet, to efficiently quantify epistemic uncertainty in PINNs. The proposed approach works as an add-on to existing, pre-trained PINNs with a small computational overhead. We demonstrate the applicability of the proposed framework in various test cases and compare the results with B-PINNs using Hamiltonian Monte Carlo (HMC) posterior estimation and dropout-equipped PINNs (Dropout-PINNs). In our experiments, E-PINNs achieve calibrated coverage with competitive sharpness at substantially lower cost. We demonstrate that when B-PINNs produce narrower bands, they under-cover in our tests. E-PINNs also show better calibration than Dropout-PINNs in these examples, indicating a favorable accuracy-efficiency trade-off.