Physics informed deep learning for computational elastodynamics without labeled data (2006.08472v1)

Abstract: Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neural network (DNN) and embed the physical law to regularize the network. To this end, training the network is equivalent to minimization of a well-designed loss function that contains the PDE residuals and initial/boundary conditions (I/BCs). In this paper, we present a physics-informed neural network (PINN) with mixed-variable output to model elastodynamics problems without resort to labeled data, in which the I/BCs are hardly imposed. In particular, both the displacement and stress components are taken as the DNN output, inspired by the hybrid finite element analysis, which largely improves the accuracy and trainability of the network. Since the conventional PINN framework augments all the residual loss components in a "soft" manner with Lagrange multipliers, the weakly imposed I/BCs cannot not be well satisfied especially when complex I/BCs are present. To overcome this issue, a composite scheme of DNNs is established based on multiple single DNNs such that the I/BCs can be satisfied forcibly in a "hard" manner. The propose PINN framework is demonstrated on several numerical elasticity examples with different I/BCs, including both static and dynamic problems as well as wave propagation in truncated domains. Results show the promise of PINN in the context of computational mechanics applications.

• The paper introduces a mixed-variable PINN that predicts both displacement and stress fields, enhancing accuracy and network trainability.
• The paper enforces initial and boundary conditions exactly by decomposing the solution into particular and general networks.
• The paper validates the framework on benchmark problems, achieving low errors compared to FEM in static and dynamic elastodynamic scenarios.

Physics-Informed Deep Learning for Elastodynamics

This paper presents a novel methodological framework employing Physics-Informed Neural Networks (PINNs) for solving computational elastodynamics problems without the need for labeled data. The approach leverages the potential of deep learning models to approximate solutions to partial differential equations (PDEs) pertinent to solid mechanics, primarily elastodynamics, utilizing the principles of neural network-based modeling.

Key Methodological Innovations

One of the significant contributions of this paper is the introduction of a mixed-variable PINN framework. Unlike traditional approaches where either displacement or stress might serve as the sole output variable, this method concurrently predicts both displacement and stress fields. This dual-target formulation is inspired by hybrid finite element schemes and is shown to enhance both the predictive accuracy and trainability of the network.

The authors introduce a composite strategy for hard enforcement of initial and boundary conditions (I/BCs). This contrasts with the traditional soft enforcement that relies on penalty terms in the loss function, which can sometimes lead to insufficient satisfaction of I/BCs. The proposed composite network achieves this by decomposing the solution into a particular solution network trained solely on I/BCs and a general solution network tasked with minimizing the PDE residuals. Such a decomposition allows exact satisfaction of the conditions, which is crucial for the uniqueness and stability of the solutions in physics-guided modeling.

Numerical Results and Validation

The framework is validated on several benchmark problems, including static and dynamic scenarios of a defected plate under tension and elastic wave propagation in confined and semi-infinite domains. Notably, the proposed PINN framework successfully captures the stress concentrations around defects and effectively predicts the wave propagation phenomena without exhibiting reflective artifacts typically mitigated by artificial boundary conditions in traditional numerical simulations.

Extensive numerical experiments demonstrate the ability of the mixed-variable PINN to achieve low errors compared to Finite Element Method (FEM) solutions. For instance, in the dynamic wave propagation scenarios, the results showcase substantial agreement in stress fields when juxtaposed with established numerical solvers, elucidating the robustness of the PINN even in complex geometries where exact solutions are sparse.

Implications and Future Directions

The methodological advancements posited by this paper have substantial implications for the computational modeling community, particularly in domains where obtaining labeled training data is cost-prohibitive or technically challenging. The framework's ability to handle both confined and unbounded problems suggests its potential applicability in geophysics, biomechanics, and materials engineering where elastodynamic analyses are prevalent.

Furthermore, the integrated approach of combining traditional physics constraints with neural networks opens avenues for further explorations into hybrid modeling methods. Potential future work could extend this framework to nonlinear mechanics, stochastic PDEs, and multiscale phenomenon modeling. The paper also hints at an inherent extensibility to incorporate stochastic variational approaches for uncertainty quantification, vital for simulations involving inherently uncertain parameters or sparse data scenarios.

In summary, this research offers a compelling stride towards embedding physical laws into deep learning frameworks, heralding a new class of PINNs for physics-based modeling across diverse scientific and engineering applications.