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Physics informed deep learning for computational elastodynamics without labeled data

Chengping Rao, Hao Sun, Yang Liu

arXiv:2006.08472v1math.NAcs.AIcs.CEcs.LG

TL;DR

The paper addresses data-free solution of elastodynamics PDEs while handling the accuracy and trainability challenges of conventional PINNs. It uses mixed displacement–stress outputs and a composite DNN scheme for hard initial/boundary-condition enforcement, and demonstrates the framework on several elasticity problems with agreement to finite-element references.

  • Problem

    Conventional data-driven physical models require rich data and may not satisfy physical constraints, while PINNs face difficulties accurately enforcing initial and boundary conditions.

  • Method

    The framework approximates displacement and stress with a DNN and uses a composite PINN construction to enforce initial and boundary conditions in a hard manner.

  • Results

    The proposed PINN improves accuracy and trainability over pure displacement-based PINNs and produces results consistent with finite-element reference solutions across elasticity examples.

  • Takeaways & Limitations

    PINNs show promise for computational elastodynamics without labeled data, including static, dynamic, and wave-propagation problems in truncated domains.

  • Takeaways & Limitations

    The formulation addresses the nontrivial enforcement of initial and boundary equality constraints through a composite scheme because soft enforcement may be inaccurate.

Abstract

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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.

1. Introduction

The paper positions physics-informed neural networks as a data-free approach for modeling PDE-governed physical systems, using deep networks to approximate physical fields while embedding governing physics. It applies this approach to elastodynamics through mixed displacement–stress outputs and forcibly imposed initial/boundary conditions.

  • 1. Introduction: Deep-learning approaches for physical modeling divide between data-driven surrogate models and PINNs that model governing PDE solutions.Surrogate models require rich data and may fail to satisfy physical constraints, whereas PINNs incorporate the governing equations into the modeling process.
  • 1. Introduction: The paper develops a PINN framework for elastodynamics, including wave propagation in bounded and truncated domains, without labeled data.A feedforward DNN maps spatiotemporal coordinates X = (x, t) to displacement and stress fields Y = (u, σ).
  • 1. Introduction: The proposed framework uses a mixed-variable output containing both displacement and stress components, motivated by hybrid finite element analysis.The paper identifies this formulation as important for improving network accuracy and trainability.
  • 1. Introduction: A composite scheme of multiple DNNs is proposed so that initial and boundary conditions can be satisfied forcibly in a hard manner.The framework is presented as a synergy solution for elastodynamics problems with varied initial and boundary conditions.
  • 1. Introduction: The paper evaluates the framework on static and dynamic elasticity examples, including a defected plate and elastic-wave propagation in confined and truncated domains.These examples are used to illustrate the capability of the proposed PINN framework for computational elastodynamics.

2. Method

The method uses DNNs as physics-informed approximators for elastodynamics, with automatic differentiation, collocation-based residual evaluation, and mixed displacement–stress outputs. A composite DNN scheme hard-enforces initial and boundary conditions, addressing inaccuracies associated with conventional soft enforcement.

  • Deep neural network and automatic differentiation: A feedforward DNN maps spatiotemporal coordinates X = (x, t) to physical quantities and is trained as a parameterized approximate solution.The network uses trainable weights and biases, while its loss embeds the governing physical law.
  • Deep neural network and automatic differentiation: Automatic differentiation computes derivatives such as ∇·σ and u_tt from the neural-network computational graph with only round-off error.These derivatives are needed to construct PDE residuals for PINN training.
  • PINN as PDE solver: PINN discretizes the spatiotemporal domain with collocation points to evaluate equation, initial-value, and boundary-value residuals.The collocation set contains points associated with Dirichlet, Neumann, and initial conditions, and can be generated using Latin hypercube sampling.
  • Elasticity theory and mixed-variable formulation: The elastodynamics formulation outputs both displacement and stress fields, which is reported to improve network trainability relative to displacement-only output.This mixed-variable formulation is inspired by hybrid finite element analysis.
  • Elasticity theory and mixed-variable formulation: Conventional PINNs impose initial and boundary conditions softly through weighted residual losses, but these conditions may not be enforced accurately.The weighting coefficients require trial and error to mitigate gradient-related pathology.
  • Elasticity theory and mixed-variable formulation: A composite scheme of three DNNs provides hard initial and boundary-condition enforcement while the remaining network is trained to satisfy governing-equation residuals.The components represent the particular solution, distance function, and general solution, respectively.

3. Results

The proposed PINN reproduces static, dynamic, and elastic-wave solutions with hard I/BC enforcement, showing agreement with finite-element references across bounded and truncated domains.

  • 3.1. Defected plate under periodic uni-axial tension: The defected-plate problem models a quarter of a 1.0 m square plate with a central 0.1 m circular defect under 1.0 MPa traction.The plate has Young’s modulus 20 MPa and Poisson’s ratio 0.25; I/BCs are imposed hard.
  • 3.1. Defected plate under periodic uni-axial tension: As network width and depth increase, both the loss and relative ℓ2 errors for von Mises stress and displacement decrease.PINN results converge with increasing total neuron count and agree satisfactorily with the FE reference fields.
  • 3.1. Defected plate under periodic uni-axial tension: The predicted static displacement and stress fields agree with the finite-element solution, including the stress distribution along the notch surface.The convergence analysis specifically examines stress on the notched surface, while field comparisons cover displacement and stress.
  • 3.1. Defected plate under periodic uni-axial tension: Under cyclic traction over 10 s, PINN stress fields remain consistent with implicit FEM and capture von Mises stress evolution at the vulnerable notch point.The cyclic load is Tn(t) = 0.5sin(2πt/T0 + 1.5π)+0.5 with T0 = 5 s.
  • 3.2. Elastic wave propagation: For confined elastic-wave propagation, the PINN predicts displacement and stress fields using hard I/BCs and a composite network architecture.The reported networks are 6 × 140 for Uh, 3 × 30 for D, and 3 × 20 for Up; the reference uses 41,696 linear quadrilateral elements with a 0.01 s time step.
  • 3.2.3. Semi-infinite domain: In the truncated semi-infinite domain, PINN captures surface reflection without interference from the other fixed edges, and its mid-line vertical displacement matches the FE solution.The comparison uses x = 0 and y ∈ [2, 15] m at various moments.

4. Conclusions

The paper develops a labeled-data-free PINN framework for elastodynamics using mixed displacement–stress outputs and composite hard enforcement of initial and boundary conditions.

  • The framework models elastodynamics problems without labeled data, while allowing measurement data when available.
  • Mixed-variable PINNs output both displacement and stress, improving accuracy and trainability over pure displacement-based PINNs.
  • A composite PINN constructs a synergy solution that imposes initial and boundary conditions forcibly rather than through weak loss penalties.
  • The framework mitigates boundary inaccuracies encountered by conventional soft enforcement, as shown by the parametric study.
  • Static and dynamic examples include a defected plate under cyclic tension and wave propagation in confined, semi-infinite, and infinite domains.

Appendix A. Comparison between mixed-variable/pure-displacement formulation

The mixed-variable formulation improves PINN training for the static plane-stress case by reducing derivative order and producing satisfactory stress fields where the pure-displacement formulation fails.

  • The mixed-variable network maps spatial coordinates to displacement and stress variables, whereas the pure-displacement network outputs only displacement.
  • Mixed-variable formulation reduces the highest spatial derivative order from two to one through the displacement–strain relationship.
  • The mixed output also enables hard imposition of Neumann conditions in addition to Dirichlet conditions, unlike the pure-displacement formulation.
  • 2.89 × 10^-1 and 2.93 × 10^-5 are the final loss values for mixed-variable and pure-displacement PINNs, respectively.
  • With identical hyperparameters and collocation points, the pure-displacement PINN fails to model the stress field, while the mixed-variable PINN gives satisfactory predictions against the FE reference.

Appendix B. Comparison between soft/hard enforcement of boundary conditions

The boundary-condition comparison shows that soft enforcement can leave substantial boundary inaccuracies, whereas the proposed hard-enforcement construction satisfies conditions forcibly and avoids loss-weight balancing issues.

  • The plane-stress test fixes the lower edge and applies forced displacement to half the top edge of a square plate.
  • The hard-enforcement configuration is compared with soft-enforced PINNs using different weighting coefficients and with a finite-element reference.
  • For soft enforcement with λ = 1, the lower-left boundary condition is not accurately enforced; increasing λ mitigates the inaccuracy.
  • Soft enforcement creates loss-term imbalance and gradient pathology, requiring computationally costly trial and error to select weighting coefficients.
  • The proposed construction imposes the boundary condition forcibly, eliminating the weighting-coefficient issue identified for soft enforcement.
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