DGNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling

• URL: http://arxiv.org/abs/2502.06250v1
• Date: Mon, 10 Feb 2025 08:28:46 GMT
• Title: DGNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling
• Authors: Yaohua Zang, Phaedon-Stelios Koutsourelakis,
• Abstract summary: Deep Generative Neural Operator (DGNO) is a physics-aware framework for solving PDE-based inverse problems. New innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems.
• Score: 1.8416014644193066
• License:
• Abstract: Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and require large amounts of {\em labeled} training data. We propose the Deep Generative Neural Operator (DGNO), a physics-aware framework that addresses these challenges by leveraging a deep, generative, probabilistic model in combination with a set of lower-dimensional, latent variables that simultaneously encode PDE-inputs and PDE-outputs. This formulation can make use of unlabeled data and significantly improves inverse problem-solving, particularly for discontinuous or discrete-valued input functions. DGNO enforces physics constraints without labeled data by incorporating as virtual observables, weak-form residuals based on compactly supported radial basis functions (CSRBFs). These relax regularity constraints and eliminate higher-order derivatives from the objective function. We also introduce MultiONet, a novel neural operator architecture, which is a more expressive generalization of the popular DeepONet that significantly enhances the approximating power of the proposed model. These innovations make DGNO particularly effective for challenging forward and inverse, PDE-based problems, such as those involving multi-phase media. Numerical experiments demonstrate that DGNO achieves higher accuracy across multiple benchmarks while exhibiting robustness to noise and strong generalization to out-of-distribution cases. Its adaptability, and the ability to handle sparse, noisy data while providing probabilistic estimates, make DGNO a powerful tool for scientific and engineering applications.

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