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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?

Physics-informed neural networks (PINNs) have become a popular choice for solving high-dimensional partial differential equations (PDEs) due to their excellent approximation power and generalization ability. Recently, extended PINNs (XPINNs) based on domain decomposition methods have attracted considerable attention due to their effectiveness in modeling multiscale and multiphysics problems and their parallelization. However, theoretical understanding of their convergence and generalization properties remains unexplored. In this study, we take an initial step towards understanding how and when XPINNs outperform PINNs. Specifically, for general multilayer PINNs and XPINNs, we first provide a prior generalization bound via the complexity of the target functions in the PDE problem and a posterior generalization bound via the posterior matrix norms of the networks after optimization. Moreover, based on our bounds, we analyze the conditions under which XPINNs improve generalization. Concretely, our theory shows that the key building block of XPINN, namely, the domain decomposition, introduces a tradeoff for generalization. On the one hand, XPINNs decompose the complex PDE solution into several simple parts, which decreases the complexity needed to learn each part and boosts generalization. On the other hand, decomposition leads to less training data being available in each subdomain, and hence such a model is typically prone to overfitting and may become less generalizable. Empirically, we choose five PDEs to show when XPINNs perform better than, similar to, or worse than PINNs, hence demonstrating and justifying our new theory.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

Here we propose a generalized space-time domain decomposition approach for the physics-informed neural networks (PINNs) to solve nonlinear partial differential equations (PDEs) on arbitrary complex-geometry domains. The proposed framework, named eXtended PINNs ( X P I N N s ), further pushes the boundaries of both PINNs as well as conservative PINNs (cPINNs), which is a recently proposed domain decomposition approach in the PINN framework tailored to conservation laws. Compared to PINN, the XPINN method has large representation and parallelization capacity due to the inherent property of deployment of multiple neural networks in the smaller subdomains. Unlike cPINN, XPINN can be extended to any type of PDEs. Moreover, the domain can be decomposed in any arbitrary way (in space and time), which is not possible in cPINN. Thus, XPINN offers both space and time parallelization, thereby reducing the training cost more effectively. In each subdomain, a separate neural network is employed with optimally selected hyperparameters, e.g., depth/width of the network, number and location of residual points, activation function, optimization method, etc. A deep network can be employed in a subdomain with complex solution, whereas a shallow neural network can be used in a subdomain with relatively simple and smooth solutions. We demonstrate the versatility of XPINN by solving both forward and inverse PDE problems, ranging from one-dimensional to three-dimensional problems, from time-dependent to time-independent problems, and from continuous to discontinuous problems, which clearly shows that the XPINN method is promising in many practical problems. The proposed XPINN method is the generalization of PINN and cPINN methods, both in terms of applicability as well as domain decomposition approach, which efficiently lends itself to parallelized computation. The XPINN code is available on h t t p s : / / g i t h u b . c o m / A m e y a J a g t a p / X P I N N s .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Adaptive Interface-PINNs (AdaI-PINNs) for inverse problems: Determining material properties for heterogeneous systems

Here, we determine spatially varying discontinuous material properties using a domain-decomposition based physics-informed neural networks (PINNs) framework named the Adaptive Interface-PINNs or AdaI-PINNs (Roy et al., 2024). We propose the use of distinct neural networks for the field variables and material properties within each material, utilizing adaptive activation functions. While the neural networks across different materials share the same weights and biases, their activation functions are uniquely tailored using a hyperparameter that influences the slope of the activation function. The proposed framework is tested on several one-dimensional and two-dimensional benchmark examples, and its performance is compared with conventional PINNs and existing domain-decomposition PINNs frameworks, namely, the Multi-domain physics-informed neural network (M-PINN), and the eXtended physics-informed neural networks (XPINNs). The results demonstrate that the proposed approach can determine randomly distributed discontinuous material properties with an L 2 error of $\mathscr{O}$ (10 -3 ) for the material property and the root-mean-square error of $\mathscr{O}$ (10 -3 ) for the primary variable while the other approaches yield errors that are approximately two orders of magnitude larger (that is, $\mathscr{O}$ (10 -1 )). Moreover, the spatial distribution of material properties obtained using the proposed framework is in close agreement with the true distribution, whereas the other approaches fare much worse. Additionally, the proposed approach is approximately 40% faster than its competitors, indicating its potential as a robust alternative for solving inverse problems in heterogeneous materials.

36 MATERIALS SCIENCE↗
DOE OSTI2021

Parallel physics-informed neural networks via domain decomposition

Here we develop a distributed framework for the physics-informed neural networks (PINNs) based on two recent extensions, namely conservative PINNs (cPINNs) and extended PINNs (XPINNs), which employ domain decomposition in space and in time-space, respectively. This domain decomposition endows cPINNs and XPINNs with several advantages over the vanilla PINNs, such as parallelization capacity, large representation capacity, efficient hyperparameter tuning, and is particularly effective for multi-scale and multi-physics problems. Here, we present a parallel algorithm for cPINNs and XPINNs constructed with a hybrid programming model described by MPI + X, where X ∈ {CPUs, GPUs}. The main advantage of cPINN and XPINN over the more classical data and model parallel approaches is the flexibility of optimizing all hyperparameters of each neural network separately in each subdomain. We compare the performance of distributed cPINNs and XPINNs for various forward problems, using both weak and strong scalings. Our results indicate that for space domain decomposition, cPINNs are more efficient in terms of communication cost but XPINNs provide greater flexibility as they can also handle time-domain decomposition for any differential equations, and can deal with any arbitrarily shaped complex subdomains. To this end, we also present an application of the parallel XPINN method for solving an inverse diffusion problem with variable conductivity on the United States map, using ten regions as subdomains.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Interface PINNs (I-PINNs): A physics-informed neural networks framework for interface problems

Here, we present a novel physics-informed neural networks (PINNs) framework for modeling interface problems, termed Interface PINNs (I-PINNs). I-PINNs uses different neural networks for any two subdomains separated by a sharp interface such that the neural networks differ only through their activation functions while the other parameters remain identical. The performance of I-PINNs, conventional PINNs, and other existing domain-decomposition PINNs methods such as extended PINNs (XPINNs) and multi-domain PINN (M-PINN) is compared through several one-dimensional, two-dimensional, and three-dimensional benchmark elliptic interface problems. The results demonstrate that I-PINNs provides a root-mean-square-error accuracy, at least two orders of magnitude better than conventional PINNs and XPINNs at approximately one-tenth of the computational cost of conventional PINNs and half the cost of XPINNs. Additionally, while I-PINNs and M-PINN provide comparable accuracies, M-PINN is found to be approximately 50% more expensive.

42 ENGINEERING↗
DOE OSTI2024

Physics-informed neural networks for heterogeneous poroelastic media

This study presents a novel physics-informed neural network (PINN) framework for modeling poroelasticity in heterogeneous media with material interfaces. The approach introduces a composite neural network (CoNN) where separate neural networks predict displacement and pressure variables for each material. While sharing identical activation functions, these networks are independently trained for all other parameters. To address challenges posed by heterogeneous material interfaces, the CoNN is integrated with the Interface-PINNs (I-PINNs) framework (Sarma et al., Comput. Methods Appl. Mech. Eng. 429: 117135, 2024), allowing different activation functions across material interfaces. Further, this ensures accurate approximation of discontinuous solution fields and gradients. Performance and accuracy of this combined architecture were evaluated against the conventional PINNs approach, a single neural network (SNN) architecture, and the eXtended PINNs (XPINNs) framework through two one-dimensional benchmark examples with discontinuous material properties. The results show that the proposed CoNN with I-PINNs architecture achieves an RMSE that is two orders of magnitude better than the conventional PINNs approach and is at least 40 times faster than the SNN framework. Compared to XPINNs, the proposed method achieves an RMSE at least one order of magnitude better and is 40% faster.

42 ENGINEERING↗