Scalable algorithms for physics-informed neural and graph networks

Abstract

Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space–time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.

Physics-Informed Deep Neural Operator Networks

“…Adding physics into the training of neural operators in addition to any available data enhances their accuracy and generalization capacity for tasks even outside the distribution of the input space. Scalable physics-informed neural networks [59] can be employed to solve high-dimensional problems not possible with traditional finite element solvers, e.g., up to approximately 10 dimensions if not more. Similarly, scalable physics-informed neural operators can also solve high-dimensional problems even in real time and can be used for designing very complex systems.…”

Fundamental Understanding of Heat and Mass Transfer Processes for Physics-Informed Machine Learning-Based Drying Modelling

“…Therefore, no domain discretisation errors or other solver-based limitations are imposed in PIML calculations. For moving boundaries or domains, PIML does not demand complex computational advancement requirements as traditional computational approaches do [106]. In addition, inherent neural network capabilities can be integrated with physics-based model solutions to overcome most of the limitations of traditional computational approaches [17,107,108].…”

“…Therefore, PIML can be an alternative approach to the traditional computational approaches where they struggle. For instance, it can extract additional information when scattered data is avail-able and conditions are ill-posed [20,106,112,113]. Cai et al [107] and Blechschmidt and Ernst [108] showed such substantial capabilities in heat transfer and high-dimensional problems, respectively.…”

Physics-informed neural networks for structural health monitoring: a case study for Kirchhoff–Love plates

“…PINNs were introduced as a new class of data-driven solvers and as a novel approach in scientific machine learning for dealing with problems involving partial differential equations (PDEs) (Cuomo et al, 2022) using automatic differentiation. Several studies have since applied PINNs for a range of scientific problems (Shukla et al, 2022), involving stochastic PDEs, integro-differential equations, fractional PDEs, non-linear differential equations (Uddin et al, 2023), and optimal control of PDEs (Mowlavi and Nabi, 2023). Within engineering, PINNs have been applied to problems in fluid dynamics (Mao et al, 2020; Raissi et al, 2020; Sliwinski and Rigas, 2023), heat transfer (Zobeiry and Humfeld, 2021), tensile membranes (Kabasi et al, 2023), and material behavior modeling (Zheng et al, 2022).…”