# Interval and fuzzy physics-informed neural networks for uncertain fields

arXiv: https://arxiv.org/abs/2106.13727 · HTML: https://mlsanity.com/paper/2106.13727/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 52 references that are in the map.
- First posted on arXiv 18 Jun 2021.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Deep learning of thermodynamics-aware reduced-order models from data](https://mlsanity.com/paper/2007.03758/) (2020)
- [A framework for data-driven solution and parameter estimation of PDEs using conditional generative adversarial networks](https://mlsanity.com/paper/2105.13136/) (2021)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [A machine learning based plasticity model using proper orthogonal decomposition](https://mlsanity.com/paper/2001.03438/) (2020)
- [Physics-informed Neural Networks for Solving Nonlinear Diffusivity and Biot's equations](https://mlsanity.com/paper/2002.08235/) (2020)
- [Local approximate Gaussian process regression for data-driven constitutive laws: Development and comparison with neural networks](https://mlsanity.com/paper/2105.04554/) (2021)
- [Non-intrusive reduced order modeling of poroelasticity of heterogeneous media based on a discontinuous Galerkin approximation](https://mlsanity.com/paper/2101.11810/) (2021)
- [Model-data-driven constitutive responses: application to a multiscale computational framework](https://mlsanity.com/paper/2104.02650/) (2021)
- [Adaptive surrogate models for parametric studies](https://mlsanity.com/paper/1905.05345/) (2019)

## Built on by (most cited first, 9 of 9)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Modular machine learning-based elastoplasticity: generalization in the context of limited data](https://mlsanity.com/paper/2210.08343/) (2022)
- [Spiking neural networks for nonlinear regression](https://mlsanity.com/paper/2210.03515/) (2022)
- [Stochastic Inference of Plate Bending from Heterogeneous Data: Physics-informed Gaussian Processes via Kirchhoff-Love Theory](https://mlsanity.com/paper/2405.12802/) (2024)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Adapting Physics-Informed Neural Networks for Bifurcation Detection in Ecological Migration Models](https://mlsanity.com/paper/2409.00651/) (2024)
- [Perception-Informed Neural Networks: Beyond Physics-Informed Neural Networks](https://mlsanity.com/paper/2505.03806/) (2025)
- [Interval and fuzzy physics-augmented neural networks (iPANN and fPANN) for uncertainty quantification and propagation in constitutive modeling](https://mlsanity.com/paper/2607.20339/) (2026)
- [Quantum-Classical Physics-Informed Kolmogorov-Arnold Networks for Solving Fuzzy Differential Equations](https://mlsanity.com/paper/2608.08782/) (2026)

## Abstract (arXiv)

Temporally and spatially dependent uncertain parameters are regularly encountered in engineering applications. Commonly these uncertainties are accounted for using random fields and processes, which require knowledge about the appearing probability distributions functions that is not readily available. In these cases non-probabilistic approaches such as interval analysis and fuzzy set theory are helpful uncertainty measures. Partial differential equations involving fuzzy and interval fields are traditionally solved using the finite element method where the input fields are sampled using some basis function expansion methods. This approach however is problematic, as it is reliant on knowledge about the spatial correlation fields. In this work we utilize physics-informed neural networks (PINNs) to solve interval and fuzzy partial differential equations. The resulting network structures termed interval physics-informed neural networks (iPINNs) and fuzzy physics-informed neural networks (fPINNs) show promising results for obtaining bounded solutions of equations involving spatially and/or temporally uncertain parameter fields. In contrast to finite element approaches, no correlation length specification of the input fields as well as no Monte-Carlo simulations are necessary. In fact, information about the input interval fields is obtained directly as a byproduct of the presented solution scheme. Furthermore, all major advantages of PINNs are retained, i.e. meshfree nature of the scheme, and ease of inverse problem set-up.