Plasolver, a physics-informed neural operator framework that combines the efficiency of operator learning with the accuracy and robustness of classical numerical solvers, provides an efficient, accurate, and discretization-invariant computational framework for nonlinear, path-dependent elastoplastic problems.
Abstract
Elastoplastic analysis is computationally demanding because its nonlinear, path-dependent constitutive behavior requires incremental loading and repeated iterative solutions. To address this challenge, we propose Plasolver, a physics-informed neural operator framework that combines the efficiency of operator learning with the accuracy and robustness of classical numerical solvers. Plasolver consists of a physics-informed pretraining stage and an optional warm-start stage. During pretraining, the neural operator is trained solely by minimizing the incremental potential energy of elastoplasticity formulated by Simo, without requiring any labeled solution data. It operates directly on unstructured point clouds by encoding spatial coordinates, loading histories, and material properties as unified point-wise prompts. This formulation provides dual invariance to spatial and loading-path discretizations, enabling consistent predictions across different spatial resolutions and different numbers of increments representing the same loading trajectory. The pretrained Plasolver achieves relative errors on the order of 1\% while providing approximately two orders of magnitude acceleration over conventional finite element simulations. In the warm-start stage, the pretrained prediction is supplied as the initial solution to a classical iterative solver, preserving its numerical accuracy, robustness, and convergence properties while substantially accelerating convergence. Numerical results show that Plasolver reduces the required number of iterations by approximately 50\% compared with conventional zero-initialized solvers and converges to solutions at any prescribed tolerance. Plasolver thus provides an efficient, accurate, and discretization-invariant computational framework for nonlinear, path-dependent elastoplastic problems.
Designing multilayered materials for controlled deformation is heavily bottlenecked by the immense computational costs and time of traditional, industrial-grade finite element methods (FEM). This excessive expense severely limits design space exploration and gradient-based optimization. To streamline the workflow, we propose a framework for parameterized physics-informed neural networks (P2INNs). The framework encompasses three core components: I. a FEM baseline with a Hex8 element and trilinear basis functions, II. a P2INN displacement field model across variable material stiffness and layer thicknesses, and III. FEM-referenced evaluation. The PINN enforces static linear elasticity via Navier-Cauchy residuals with training strategies that prioritize physics, including layerwise PDE decomposition, interface continuity penalty, and compliance aware scaling. For one-layer, pure physics-driven configurations, the PINN achieves a mean volume MAE of 1.56% and worst-case volume MAE of 2.87% against the FEM benchmark. For three-layer configurations with controlled supervised training, the PINN achieves a mean volume MAE of 2.53% and worst-case volume MAE of 4.66%, meeting the near-5% worst-case volume-MAE target for preliminary design-space exploration. The trained P2INN model remains a lightweight model with a simple forward pass for future calculations, creating an alternative to traditional FEM. By drastically reducing the need for expensive FEM evaluations, this approach could accelerate the inverse design and optimization of advanced layered architectures for protective structures in various load-heavy or potentially collision-heavy fields.
Characterizing the mechanical response of materials traditionally requires solving optimization problems in which model parameters are calibrated or trained to minimize the discrepancy between model predictions and experimental data. This process can be computationally expensive and time-consuming. To overcome this limitation, we propose two neural operator architectures that directly map experimentally measured data to the constitutive functions governing the mechanical response of the material: Physics-Augmented Neural Operators (PANO) and Constitutive Artificial Neural Operators (CANO). The proposed neural operators approximate the mapping between the infinite-dimensional input space of full-field displacement measurements and net reaction forces, and the infinite-dimensional output space of hyperelastic strain-energy density functions. The displacement fields are encoded through Laplacian eigenfunctions to obtain discretization-independent and noise-robust predictions. Our framework constrains the output space to physically admissible material models that satisfy fundamental physical requirements by design. The neural operators are trained on simulated data tuples of displacement fields and reaction forces for a range of material models. Once trained, the neural operators enable near-instantaneous material characterization and require only a single forward pass to infer the strain-energy density function from a given experimental dataset. We test the predictive power of the neural operators for unseen data, noisy data, data with missing information, data from different spatial discretizations, and data from geometries of different sizes.
Moritz Flaschel, Burigede Liu, Ellen Kuhl· arXiv.org· 1 citation
Physics-Informed Neural Networks (PINNs) serve as continuous, mesh-free solvers for partial differential equations, but they frequently encounter optimization failures when applied to strongly coupled, stiff multiphysics systems. In piezoelectricity, the disparity in energetic scales between mechanical stress and electric displacement creates severely ill-conditioned loss landscapes, resulting in gradient pathologies. Standard PINN formulations rely on penalty-based soft constraints for boundary and initial conditions, which exacerbate this stiffness, leak unphysical energy, and corrupt boundary stress calculations. In this work, we present a constrained PINN architecture for 1D coupled electro-elastodynamics that structurally bypasses this gradient competition. By applying the quasi-static approximation and utilizing analytical distance functions, we enforce exact Dirichlet boundaries and second-order time initial conditions directly within the neural network’s topology. This geometric constraint restricts the optimizer to a physically consistent energy manifold and enables recovery of maximal boundary stresses without internal gradient noise, a capability that is critical for predicting mechanical fatigue and failure in high-frequency transducer architectures. Stabilized by a deterministic quasi-Newton optimization stage, our fully constrained model resolves the mechanical displacement and electric potential fields with a global relative L2 error of O(10−5). These results demonstrate that the gradient pathologies typically observed in stiff multiphysics PINNs can be systematically neutralized through the exact algebraic imposition of boundary and initial constraints.
Suhas Suresh Bharadwaj· Journal of Applied Physics· 0 citations
The demand for high-fidelity, physically-based animation has traditionally been met by sophisticated solvers rooted in elastodynamics and finite element analysis (FEM). Recently, the emergence of neural physics has led to a paradigm shift, transforming neural networks into solvers with memory that dramatically increase the scale and speed of digital environments. Despite its reputation, physics-based simulation does not have to be intimidating. This course aims to demystify the field, proving that these complex systems are accessible and intuitive when approached correctly. We provide a unified journey from classical formulations to modern neural techniques, grounding the audience in the fundamentals of elastostatics and dynamics. We demonstrate how physical problems are discretized via linear finite elements and solved through the elegant lens of optimization. Transitioning into neural physics, we showcase how traditional simulation knowledge translates directly into machine learning loss functions and neural architectures. We analyze strategies for modeling latent spaces for a system’s equilibrium states and to create truly controllable, real-time frameworks. Designed for a broad audience—including students, engineers, researchers, and artists—this course balances theory with practice. To ensure these concepts are immediately actionable, we provide comprehensive reference code for all discussed methods. By the end of the session, attendees will possess the tools to quickly and easily implement their own physics solvers, empowering them to build the next generation of physics-enhanced frameworks and interactive worlds.
D. Corigliano, Otman Benchekroun, J. Barbič· Proceedings of the Special I...· 0 citations
Parametric nonlinear solid-mechanics simulations are widely used in virtual testing, optimisation, and uncertainty analysis, but repeated finite-element simulations become costly when geometry changes. This paper presents PI-GINOT, a physics-informed neural operator that predicts finite-strain hyperelastic responses across a four-parameter family of DogBone specimens without using finite-element training data. Each specimen is described by a boundary point cloud, which is encoded into geometry features. A cross-attention decoder then predicts displacement at arbitrary points. Displacement boundary conditions are enforced exactly, while stresses are computed using automatic differentiation and a compressible Neo-Hookean plane-stress model. Training is guided by equilibrium, traction-free and symmetry conditions, deformation stability, and internal force consistency. Abaqus simulations are used only after training for validation. Across eight test geometries, PI-GINOT achieves displacement errors of 2.1%-7.1%, peak von Mises stress errors of 0.9%-13.3%, and section-force errors below 10.3%. Larger errors occur in individual stress components, especially for narrow specimens, mainly because of steep stress gradients near the gauge-to-fillet transition. These results show that PI-GINOT can provide useful geometry-dependent predictions for nonlinear solid mechanics without labelled simulation data, while also revealing where better local stress resolution is still needed.
We propose an unsupervised learning framework for calibrating a physics-augmented neural network (PANN) for small-strain viscoelasticity via full-field data. It only requires quantities that are directly accessible in real experiments for training, namely global reaction forces and surface displacements. The underlying PANN is embedded in the generalized standard materials theory, in which two scalar-valued potentials render the constitutive model thermodynamically consistent by construction, while invariant-based representations of the free energy and the dual dissipation potential additionally ensure material symmetry. Considering a thin specimen under the plane stress assumption, we formulate a constrained optimization problem based on the equilibrium gap method in combination with quasi-Newton optimizers and automatic differentiation. Thereby, the unknown out-of-plane strain follows from the plane stress condition and the evolution of the internal variables is captured by an implicit time integration scheme. The resulting system of nonlinear equations is solved via a local Newton iteration at quadrature point and time step. To drastically reduce the computational cost of training, the backward adjoint method is employed to compute the gradient of the target loss, instead of backpropagating through all Newton iteration steps. The proposed framework is demonstrated for synthetic data, including noisy displacements and forces, showing excellent agreement across a wide range of deformation rates and load paths.
Brain M. Riemer, Markus Kästner, Karl A. Kalina· 0 citations
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