Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems

• URL: http://arxiv.org/abs/2412.03161v2
• Date: Fri, 07 Feb 2025 09:56:51 GMT
• Title: Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems
• Authors: Sung Woong Cho, Hwijae Son,
• Abstract summary: Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities. Existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. We propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs) which can learn the solution operator of PDE-based inverse problems without labeled training data.
• Score: 1.9490282165104331
• License:
• Abstract: Inverse problems involving partial differential equations (PDEs) can be seen as discovering a mapping from measurement data to unknown quantities, often framed within an operator learning approach. However, existing methods typically rely on large amounts of labeled training data, which is impractical for most real-world applications. Moreover, these supervised models may fail to capture the underlying physical principles accurately. To address these limitations, we propose a novel architecture called Physics-Informed Deep Inverse Operator Networks (PI-DIONs), which can learn the solution operator of PDE-based inverse problems without labeled training data. We extend the stability estimates established in the inverse problem literature to the operator learning framework, thereby providing a robust theoretical foundation for our method. These estimates guarantee that the proposed model, trained on a finite sample and grid, generalizes effectively across the entire domain and function space. Extensive experiments are conducted to demonstrate that PI-DIONs can effectively and accurately learn the solution operators of the inverse problems without the need for labeled data.

Related papers

• Diffeomorphic Latent Neural Operators for Data-Efficient Learning of Solutions to Partial Differential Equations [5.308435208832696]
  A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. We propose that in order to learn a PDE solution operator that can generalize across multiple domains without needing to sample enough data expressive enough, we can train instead a latent neural operator on just a few ground truth solution fields.
  arXiv  Detail & Related papers  (2024-11-27T03:16:00Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• DeltaPhi: Learning Physical Trajectory Residual for PDE Solving [54.13671100638092]
  We propose and formulate the Physical Trajectory Residual Learning (DeltaPhi) We learn the surrogate model for the residual operator mapping based on existing neural operator networks. We conclude that, compared to direct learning, physical residual learning is preferred for PDE solving.
  arXiv  Detail & Related papers  (2024-06-14T07:45:07Z)
• Solving Partial Differential Equations in Different Domains by Operator Learning method Based on Boundary Integral Equations [13.495279709392104]
  This article explores operator learning models that can deduce solutions to partial differential equations (PDEs) on arbitrary domains without requiring retraining. We introduce two innovative models rooted in boundary integral equations (BIEs) Once fully trained, these BIE-based models adeptly predict the solutions of PDEs in any domain without the need for additional training.
  arXiv  Detail & Related papers  (2024-06-04T13:19:06Z)
• Score-based Diffusion Models in Function Space [137.70916238028306]
  Diffusion models have recently emerged as a powerful framework for generative modeling. This work introduces a mathematically rigorous framework called Denoising Diffusion Operators (DDOs) for training diffusion models in function space. We show that the corresponding discretized algorithm generates accurate samples at a fixed cost independent of the data resolution.
  arXiv  Detail & Related papers  (2023-02-14T23:50:53Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Neural Inverse Operators for Solving PDE Inverse Problems [5.735035463793008]
  We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.
  arXiv  Detail & Related papers  (2023-01-26T15:12:58Z)
• Semi-supervised Invertible DeepONets for Bayesian Inverse Problems [8.594140167290098]
  DeepONets offer a powerful, data-driven tool for solving parametric PDEs by learning operators. In this work, we employ physics-informed DeepONets in the context of high-dimensional, Bayesian inverse problems.
  arXiv  Detail & Related papers  (2022-09-06T18:55:06Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Characterizing possible failure modes in physics-informed neural networks [55.83255669840384]
  Recent work in scientific machine learning has developed so-called physics-informed neural network (PINN) models. We demonstrate that, while existing PINN methodologies can learn good models for relatively trivial problems, they can easily fail to learn relevant physical phenomena even for simple PDEs. We show that these possible failure modes are not due to the lack of expressivity in the NN architecture, but that the PINN's setup makes the loss landscape very hard to optimize.
  arXiv  Detail & Related papers  (2021-09-02T16:06:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.