# Physics-constrained polynomial chaos expansion for scientific machine learning and uncertainty quantification

arXiv: https://arxiv.org/abs/2402.15115 · HTML: https://mlsanity.com/paper/2402.15115/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 51 references that are in the map.
- First posted on arXiv 23 Feb 2024.

## Builds on (most cited first, 15 of 15)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [A Dual-Dimer Method for Training Physics-Constrained Neural Networks with Minimax Architecture](https://mlsanity.com/paper/2005.00615/) (2020)
- [Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark](https://mlsanity.com/paper/2002.01290/) (2020)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Data-driven polynomial chaos expansion for machine learning regression](https://mlsanity.com/paper/1808.03216/) (2018)
- [Physics-Informed Polynomial Chaos Expansions](https://mlsanity.com/paper/2309.01697/) (2023)
- [UQpy v4.1: Uncertainty Quantification with Python](https://mlsanity.com/paper/2305.09572/) (2023)
- [Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion](https://mlsanity.com/paper/2301.13635/) (2023)
- [Learning thermodynamically constrained equations of state with uncertainty](https://mlsanity.com/paper/2306.17004/) (2023)

## Built on by (most cited first, 17 of 17)

- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [Polynomial Chaos Expansion for Operator Learning](https://mlsanity.com/paper/2508.20886/) (2025)
- [Expanding the Chaos: Neural Operator for Stochastic (Partial) Differential Equations](https://mlsanity.com/paper/2601.01021/) (2026)
- [On Fractional Moment Estimation from Polynomial Chaos Expansion](https://mlsanity.com/paper/2403.01948/) (2024)
- [Response Estimation and System Identification of Dynamical Systems via Physics-Informed Neural Networks](https://mlsanity.com/paper/2410.01340/) (2024)
- [E-PINNs: Epistemic Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.19333/) (2025)
- [Adaptive Machine Learning-Driven Multi-Fidelity Stratified Sampling for Failure Analysis of Nonlinear Stochastic Systems](https://mlsanity.com/paper/2508.00734/) (2025)
- [Deep operator network for surrogate modeling of poroelasticity with random permeability fields](https://mlsanity.com/paper/2509.11966/) (2025)
- [A generative adversarial network optimization method for damage detection and digital twinning by deep AI fault learning: Z24 Bridge structural health monitoring benchmark validation](https://mlsanity.com/paper/2511.00099/) (2025)
- [A Resolution Independent Neural Operator](https://mlsanity.com/paper/2407.13010/) (2024)
- [Uncertainty Quantification for Machine Learning-Based Prediction: A Polynomial Chaos Expansion Approach for Joint Model and Input Uncertainty Propagation](https://mlsanity.com/paper/2507.14782/) (2025)
- [Conformalized Polynomial Chaos Expansion for Uncertainty-aware Surrogate Modeling](https://mlsanity.com/paper/2510.22375/) (2025)
- [CODE: A global approach to ODE dynamics learning](https://mlsanity.com/paper/2511.15619/) (2025)
- [Physics-informed Polynomial Chaos Expansion with Enhanced Constrained Optimization Solver and D-optimal Sampling](https://mlsanity.com/paper/2512.10873/) (2025)
- [Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models](https://mlsanity.com/paper/2601.03367/) (2026)
- [Uncertainty Quantification of Engineering Structures by Polynomial Chaos Expansion and Multivariate Active Learning](https://mlsanity.com/paper/2606.17233/) (2026)
- [Structured Neural Chaos: An Adaptive Surrogate Modeling Framework for Functional Uncertainty Quantification and Global Sensitivity Analysis](https://mlsanity.com/paper/2607.28903/) (2026)

## Abstract (arXiv)

We present a novel physics-constrained polynomial chaos expansion as a surrogate modeling method capable of performing both scientific machine learning (SciML) and uncertainty quantification (UQ) tasks. The proposed method possesses a unique capability: it seamlessly integrates SciML into UQ and vice versa, which allows it to quantify the uncertainties in SciML tasks effectively and leverage SciML for improved uncertainty assessment during UQ-related tasks. The proposed surrogate model can effectively incorporate a variety of physical constraints, such as governing partial differential equations (PDEs) with associated initial and boundary conditions constraints, inequality-type constraints (e.g., monotonicity, convexity, non-negativity, among others), and additional a priori information in the training process to supplement limited data. This ensures physically realistic predictions and significantly reduces the need for expensive computational model evaluations to train the surrogate model. Furthermore, the proposed method has a built-in uncertainty quantification (UQ) feature to efficiently estimate output uncertainties. To demonstrate the effectiveness of the proposed method, we apply it to a diverse set of problems, including linear/non-linear PDEs with deterministic and stochastic parameters, data-driven surrogate modeling of a complex physical system, and UQ of a stochastic system with parameters modeled as random fields.