# Calibrated Physics-Informed Uncertainty Quantification

arXiv: https://arxiv.org/abs/2502.04406 · HTML: https://mlsanity.com/paper/2502.04406/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 86 references that are in the map.
- First posted on arXiv 6 Feb 2025.

## Builds on (most cited first, 30 of 36)

- [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)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Neural General Circulation Models for Weather and Climate](https://mlsanity.com/paper/2311.07222/) (2023)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [MeshfreeFlowNet: A Physics-Constrained Deep Continuous Space-Time Super-Resolution Framework](https://mlsanity.com/paper/2005.01463/) (2020)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Predictive Inference with Feature Conformal Prediction](https://mlsanity.com/paper/2210.00173/) (2022)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Conformal Prediction Bands for Multivariate Functional Data](https://mlsanity.com/paper/2106.01792/) (2021)
- [Plasma Surrogate Modelling using Fourier Neural Operators](https://mlsanity.com/paper/2311.05967/) (2023)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Scaling physics-informed hard constraints with mixture-of-experts](https://mlsanity.com/paper/2402.13412/) (2024)
- [Active Learning for Neural PDE Solvers](https://mlsanity.com/paper/2408.01536/) (2024)
- [The Importance of Being a Band: Finite-Sample Exact Distribution-Free Prediction Sets for Functional Data](https://mlsanity.com/paper/2102.06746/) (2021)
- [Effects of boundary conditions in fully convolutional networks for learning spatio-temporal dynamics](https://mlsanity.com/paper/2106.11160/) (2021)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [Loss Landscape Engineering via Data Regulation on PINNs](https://mlsanity.com/paper/2205.07843/) (2022)
- [Conformal Methods for Quantifying Uncertainty in Spatiotemporal Data: A Survey](https://mlsanity.com/paper/2209.03580/) (2022)
- [Uncertainty Quantification of Surrogate Models using Conformal Prediction](https://mlsanity.com/paper/2408.09881/) (2024)
- [Pricing options and computing implied volatilities using neural networks](https://mlsanity.com/paper/1901.08943/) (2019)
- [Fast Dynamic 1D Simulation of Divertor Plasmas with Neural PDE Surrogates](https://mlsanity.com/paper/2305.18944/) (2023)
- [Guaranteed prediction sets for functional surrogate models](https://mlsanity.com/paper/2501.18426/) (2025)

## Built on by (most cited first, 8 of 8)

- [Plasma State Monitoring and Disruption Characterization using Multimodal VAEs](https://mlsanity.com/paper/2504.17710/) (2025)
- [Learning Physical Operators using Neural Operators](https://mlsanity.com/paper/2602.23113/) (2026)
- [Operator learning for the 2D incompressible Navier-Stokes equations: a conformal prediction approach in the data-scarce regime](https://mlsanity.com/paper/2606.08654/) (2026)
- [Distribution-Free Robust Predict-Then-Optimize in Function Spaces](https://mlsanity.com/paper/2602.08215/) (2026)
- [Data-Efficient Neural Operator Training via Physics-Based Active Learning](https://mlsanity.com/paper/2605.21348/) (2026)
- [Hybrid Neural World Models](https://mlsanity.com/paper/2605.28317/) (2026)
- [Conformal Uncertainty Quantification Guarantees for Neural Operators](https://mlsanity.com/paper/2608.28515/) (2026)
- [Predictive Uncertainty for Neural CAE Surrogates](https://mlsanity.com/paper/2609.25430/) (2026)

## Abstract (arXiv)

Simulating complex physical systems is crucial for understanding and predicting phenomena across diverse fields, such as fluid dynamics and heat transfer, as well as plasma physics and structural mechanics. Traditional approaches rely on solving partial differential equations (PDEs) using numerical methods, which are computationally expensive and often prohibitively slow for real-time applications or large-scale simulations. Neural PDEs have emerged as efficient alternatives to these costly numerical solvers, offering significant computational speed-ups. However, their lack of robust uncertainty quantification (UQ) limits deployment in critical applications. We introduce a model-agnostic, physics-informed conformal prediction (CP) framework that provides guaranteed uncertainty estimates without requiring labelled data. By utilising a physics-based approach, we can quantify and calibrate the model's inconsistencies with the physics rather than the uncertainty arising from the data. Our approach utilises convolutional layers as finite-difference stencils and leverages physics residual errors as nonconformity scores, enabling data-free UQ with marginal and joint coverage guarantees across prediction domains for a range of complex PDEs. We further validate the efficacy of our method on neural PDE models for plasma modelling and shot design in fusion reactors.