[Paper Review] Mixed formulation of physics-informed neural networks for thermo-mechanically coupled systems and heterogeneous domains

Ali Harandi, Ahmad Moeineddin+3 more|arXiv (Cornell University)|Feb 9, 2023

Model Reduction and Neural NetworksPhysics and Astronomy70 references8 citations

TL;DR

The paper develops a mixed formulation of physics-informed neural networks (PINNs) for stationary thermo-mechanical problems in heterogeneous solids, comparing sequential and coupled training, and introducing hard constraints and parametric learning to improve accuracy and efficiency.

ABSTRACT

Physics-informed neural networks (PINNs) are a new tool for solving boundary value problems by defining loss functions of neural networks based on governing equations, boundary conditions, and initial conditions. Recent investigations have shown that when designing loss functions for many engineering problems, using first-order derivatives and combining equations from both strong and weak forms can lead to much better accuracy, especially when there are heterogeneity and variable jumps in the domain. This new approach is called the mixed formulation for PINNs, which takes ideas from the mixed finite element method. In this method, the PDE is reformulated as a system of equations where the primary unknowns are the fluxes or gradients of the solution, and the secondary unknowns are the solution itself. In this work, we propose applying the mixed formulation to solve multi-physical problems, specifically a stationary thermo-mechanically coupled system of equations. Additionally, we discuss both sequential and fully coupled unsupervised training and compare their accuracy and computational cost. To improve the accuracy of the network, we incorporate hard boundary constraints to ensure valid predictions. We then investigate how different optimizers and architectures affect accuracy and efficiency. Finally, we introduce a simple approach for parametric learning that is similar to transfer learning. This approach combines data and physics to address the limitations of PINNs regarding computational cost and improves the network's ability to predict the response of the system for unseen cases. The outcomes of this work will be useful for many other engineering applications where deep learning is employed on multiple coupled systems of equations for fast and reliable computations.

Motivation & Objective

• Motivate and address solving thermo-elasticity with heterogeneous material properties using PINNs.
• Introduce a mixed PINN formulation that uses first-order derivatives and a combination of strong and weak forms.
• Compare sequential and fully coupled training strategies in terms of accuracy and computational cost.
• Incorporate hard boundary constraints to improve prediction validity and robustness.
• Propose a lightweight parametric learning approach to enhance generalization to unseen material configurations.

Proposed method

• Formulate stationary thermo-elasticity with spatially varying elastic constants and thermal properties and derive corresponding weak forms.
• Adopt a mixed PINN architecture that predicts primary fields (displacements, temperature) and their gradients/stresses via separate neural networks.
• Define physics-based loss terms combining energy-based and balance equations for mechanical and thermal fields, including boundary and connection losses.
• Implement two training strategies: sequential (alternate minimization of thermal and mechanical losses) and coupled (simultaneous minimization of all losses).
• Use first-order derivatives only to compute required quantities, enabling use of transfer-like parametric learning and hard boundary constraints to enforce physical validity.
• Incorporate hard constraints and a connection loss to link predicted stresses and heat flux to their constitutive counterparts to avoid saddle-point issues.

Experimental results

Research questions

• RQ1 Can a mixed PINN formulation using first-order derivatives improve accuracy for thermo-mechanically coupled problems in heterogeneous domains compared with standard PINNs?
• RQ2 How do sequential and coupled training strategies compare in terms of accuracy and computational cost for multi-physics PINNs?
• RQ3 What is the impact of hard boundary constraints and network architecture on the physical validity and robustness of predictions?
• RQ4 Can a parametric learning approach (transfer-learning-like) extend predictions to unseen material properties with reduced computational cost?

Key findings

• Proposes a mixed PINN framework that solves coupled thermo-elastic problems by predicting displacements, temperatures, and their associated fluxes/stresses with dedicated sub-networks.
• Demonstrates two training schemes—sequential and coupled—and discusses their accuracy and computational trade-offs.
• Shows how first-order formulations and hard constraints can improve convergence and prediction validity in heterogeneous domains.
• Introduces a simple parametric learning approach to generalize to unseen material configurations while mitigating computational cost.
• Highlights that material heterogeneity (spatially varying E, ν, k, α) is handled by including material parameters in the network input, enabling learning across configurations.