# DPA-WNO: A gray box model for a class of stochastic mechanics problem

arXiv: https://arxiv.org/abs/2309.15128 · HTML: https://mlsanity.com/paper/2309.15128/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 58 references that are in the map.
- First posted on arXiv 24 Sep 2023.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [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)
- [Transfer learning enhanced physics informed neural network for phase-field modeling of fracture](https://mlsanity.com/paper/1907.02531/) (2019)
- [Transfer learning based multi-fidelity physics informed deep neural network](https://mlsanity.com/paper/2005.10614/) (2020)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method](https://mlsanity.com/paper/2206.13103/) (2022)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Stochastic projection based approach for gradient free physics informed learning](https://mlsanity.com/paper/2209.13724/) (2022)
- [Multi-fidelity wavelet neural operator with application to uncertainty quantification](https://mlsanity.com/paper/2208.05606/) (2022)
- [Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems](https://mlsanity.com/paper/2109.00538/) (2021)
- [Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations](https://mlsanity.com/paper/2209.09750/) (2022)
- [Uncertainty Quantification and Sensitivity analysis for Digital Twin Enabling Technology: Application for BISON Fuel Performance Code](https://mlsanity.com/paper/2210.07541/) (2022)

## Built on by (most cited first, 2 of 2)

- [Generative flow induced neural architecture search: Towards discovering optimal architecture in wavelet neural operator](https://mlsanity.com/paper/2405.06910/) (2024)
- [Harnessing physics-informed operators for high-dimensional reliability analysis problems](https://mlsanity.com/paper/2409.04708/) (2024)

## Abstract (arXiv)

The well-known governing physics in science and engineering is often based on certain assumptions and approximations. Therefore, analyses and designs carried out based on these equations are also approximate. The emergence of data-driven models has, to a certain degree, addressed this challenge; however, the purely data-driven models often (a) lack interpretability, (b) are data-hungry, and (c) do not generalize beyond the training window. Operator learning has recently been proposed as a potential alternative to address the aforementioned challenges; however, the challenges are still persistent. We here argue that one of the possible solutions resides in data-physics fusion, where the data-driven model is used to correct/identify the missing physics. To that end, we propose a novel Differentiable Physics Augmented Wavelet Neural Operator (DPA-WNO). The proposed DPA-WNO blends a differentiable physics solver with the Wavelet Neural Operator (WNO), where the role of WNO is to model the missing physics. This empowers the proposed framework to exploit the capability of WNO to learn from data while retaining the interpretability and generalizability associated with physics-based solvers. We illustrate the applicability of the proposed approach in solving time-dependent uncertainty quantification problems due to randomness in the initial condition. Four benchmark uncertainty quantification and reliability analysis examples from various fields of science and engineering are solved using the proposed approach. The results presented illustrate interesting features of the proposed approach.