differentiable inverse parameter estimation

Many partial differential equation (PDE) models in science and engineering suffer from inaccurate or non-generalizable predictions due to unknown constitutive relations—e.g., nonlinear stress–strain laws or temperature-dependent thermal conductivity. To address this, we propose a fully differentiable finite element machine learning framework that embeds trainable physical operators into a general-purpose finite element method (FEM) solver, enabling end-to-end gradient propagation while preserving variational consistency. Our approach employs an encode-process-decode neural network architecture that directly models the unknown physical mapping on FEM degrees of freedom and jointly optimizes the neural operator with the FEM solver. Experiments demonstrate high-fidelity inversion of nonlinear constitutive laws from sparse experimental data and successful transfer to unseen geometries and boundary conditions in novel mechanical and thermal scenarios. The framework significantly enhances PDE models’ physical interpretability, predictive accuracy, and cross-scenario generalization capability.

This study addresses the challenge of efficiently and accurately inverting finite-strain anisotropic elastoplastic constitutive models in high-dimensional parameter spaces. To this end, we propose a JAX-based differentiable, GPU-accelerated finite element framework that tightly integrates automatic differentiation with finite element computations, eliminating the need for manual gradient derivation and enabling PDE-constrained inverse parameter identification. By leveraging heterogeneous specimen geometries inspired by topology optimization and full-field displacement measurements, the approach substantially reduces experimental dependency. Implemented on a single H100 GPU, the framework achieves up to 9.4× speedup over an Abaqus implementation running on a 24-core CPU and successfully recovers both homogeneous and spatially varying anisotropic yield and hardening parameters, demonstrating its efficiency and feasibility for high-dimensional inverse problems.

This study investigates the feasibility and geometric dependence of inferring the relative magnitudes of tensile, bending, and bearing loads from stress intensity factor (SIF) distributions along a crack front. Leveraging finite element data from SIFBench, the authors propose a unified crack-front operator that couples a structured forward surrogate model with a differentiable inverse mapping, yielding a set-valued estimator augmented with calibrated uncertainty quantification. Theoretical analysis reveals that load identifiability hinges on the functional linear independence of three canonical load profiles, and introduces an intrinsic stability margin to quantify the ill-posedness of the inverse problem. Experimental validation demonstrates that most corner-crack configurations are well-posed, yielding reliable point estimates, whereas a few inherently ill-conditioned cases produce uninformative estimates—consistent with theoretical predictions and numerical observations.

This work addresses the limitations of traditional phase-field fracture simulations, which rely on global sparse matrix assembly and thus hinder efficient GPU utilization and compatibility with automatic differentiation, impeding integration with optimization and machine learning. The authors propose the first purely PyTorch-based, matrix-free, and differentiable phase-field fracture solver, formulated along an explicit dynamics pathway. By recasting finite element operations as tensor contractions and scatter-add accumulations, the method avoids assembling the global stiffness matrix entirely. It supports multiple energy decompositions (e.g., spectral) and fracture models (AT1/AT2), operates seamlessly on both CPU and GPU, and exhibits memory consumption independent of conjugate gradient iterations. Custom backward propagation enables implicit differentiation through implicitly solved damage fields. The solver accurately reproduces six canonical fracture benchmarks on meshes with up to one million nodes and achieves end-to-end inversion of the critical fracture energy \( G_c \) with relative errors below \( 10^{-3} \).

To address insufficient calibration accuracy of elastoplastic constitutive model parameters, this work formulates a nonlinear optimization problem constrained by the constitutive evolution equations, aiming to minimize the discrepancy between predicted and experimental stress responses. Methodologically, we propose a novel solution framework integrating automatic differentiation, direct-adjoint sensitivity analysis, and a second-order Newton method—enabling, for the first time, efficient and high-accuracy analytical computation of the Hessian matrix. This overcomes convergence limitations inherent in conventional first-order algorithms (e.g., L-BFGS-B) when calibrating strongly nonlinear material models. Validation across multiple representative plasticity benchmarks demonstrates that the proposed approach significantly improves parameter identification accuracy, reduces Newton iterations by 40–60%, and markedly enhances convergence robustness. The method thus establishes a scalable, high-fidelity numerical foundation for calibrating complex constitutive models.

This study addresses the challenge that GPU scheduling, long-horizon simulations, and objective formulation in differentiable physics optimization for robotic material manipulation frequently yield numerically unreliable gradients. To investigate this, we construct a differentiable simulation benchmark based on the Material Point Method (MPM), revealing the perturbation mechanisms through which thread scheduling affects gradient computation. Furthermore, we establish a finite-difference validation criterion grounded in variance across repeated runs and propose a reproducible accumulation strategy. Our analysis confirms the risk of gradient sign reversal over extended horizons. Ultimately, this work provides concrete practical guidelines for enhancing the reliability and reproducibility of diverse differentiable simulators in robotic trajectory optimization tasks.

This work addresses the need for real-time structural health monitoring of cracked elastic bodies by proposing a physics-informed Deep Operator Network (DeepONet) that predicts linear elastic displacement fields directly from boundary conditions and crack geometry without relying on simulation data. The approach introduces a crack-geometry-specific encoding strategy and incorporates a weak-form local penalty term into the loss function to enforce traction-free boundary conditions, enabling unsupervised learning under physically consistent constraints. Validated across diverse crack configurations, the method achieves both high-fidelity predictions and real-time inference, offering a novel paradigm for surrogate modeling of complex cracked systems.

This study addresses the PDE-constrained inverse problem of correcting Reynolds-averaged Navier–Stokes (RANS) turbulence models by introducing an end-to-end differentiable framework. The approach embeds a PDE solver into PyTorch’s automatic differentiation graph via implicit layers and incorporates a trainable additive correction term, enabling joint optimization of model parameters or neural networks. Built upon the BROADCAST solver, this work presents a unified and user-friendly differentiable PDE interface and, for the first time, achieves end-to-end differentiable correction for compressible-flow RANS models. The framework successfully optimizes the production-term coefficient in the Spalart–Allmaras model and reconstructs the eddy viscosity field in two canonical cases—NASA’s wall-mounted hump and the VKI LS-59 turbine blade—demonstrating its effectiveness and flexibility for turbulence modeling and broader physics-informed PDE inverse problems.

This work addresses the challenges in inverse material design posed by discrete parameters, physical constraints, and solution multiplicity, which often render gradient-based optimization ineffective. To overcome these limitations, the authors propose a differentiable framework that integrates continuous relaxation with guided diffusion. By relaxing the discrete design space into a continuous grid and combining differentiable finite element simulation with implicit differentiation, the method leverages a diffusion model during inference, steered by a multi-objective loss function to generate physically plausible designs. This approach achieves the first integration of implicit differentiation and diffusion priors for multimodal inverse material design. It efficiently produces diverse, physically valid 2D and 3D structures with relative errors below 1%, while simultaneously optimizing multiple performance objectives such as material density.

To address the divergence issues commonly encountered in alternating minimization of non-convex energy functionals within variational phase-field models for brittle fracture, this paper proposes a robust numerical framework incorporating exact line search. The method employs a bisection-based exact line search strategy to guarantee global convergence of Newton iterations in each subproblem; it further integrates strain-energy decomposition with an irreversibility constraint to enhance physical consistency. Evaluated on standard two- and three-dimensional benchmark problems, the proposed approach demonstrates superior convergence stability and computational robustness compared to conventional line-search techniques (e.g., Armijo’s rule), particularly under challenging conditions such as strong material nonlinearity and large load increments. The key contributions include: (i) a globally convergent, exact line search tailored for phase-field fracture solvers; (ii) a physically consistent treatment of energy splitting and irreversibility; and (iii) substantial improvements in reliability across a wide range of demanding simulation scenarios.