Differentiable finite element method with Galerkin discretization for fast and accurate inverse analysis of multidimensional heterogeneous engineering structures - Wang, Xi; Yin, Zhen-Yu; Wu, Wei; Zhu, He-Hua (Computer methods in applied mechanics and engineering, 2025)
•Galerkin discretization is utilized to build a novel differentiable finite element method (DFEM) that encodes physics and significantly reduces training cost.•DFEM embeds the weak-form physics, boundary/initial conditions, and data constraints into the network architecture.•Both the accuracy and efficiency of inverse analysis are improved by several orders of magnitude.•Inverse analysis of three-dimensional heterogeneous engineering structures can be accomplished in seconds.•DFEM can be readily extended as Physics-Encoded Numerical Network (PENN) to revitalize classical numerical methods for AI4Science. Physics-informed neural networks (PINNs) are well-regarded for their capabilities in inverse analysis. However, efficient convergence is hard to achieve due to the necessity of simultaneously handling physics constraints, data constraints, blackbox weights, and blackbox biases. Consequently, PINNs are highly challenged in the inverse analysis of unknown boundary loadings and heterogeneous material parameters, particularly for three-dimensional engineering structures. To address these limitations, this study develops a novel differentiable finite element method (DFEM) based on Galerkin discretization for diverse inverse analysis. The proposed DFEM directly embeds the weak form of the partial differential equation into a discretized and differentiable computational graph, yielding a loss function from fully interpretable trainable parameters. Moreover, the labeled data, including boundary conditions, are strictly encoded into the computational graph without additional training. Finally, two benchmarks validate the DFEM's superior efficiency and accuracy: (1) With only 0.3 % training iterations, the DFEM can achieve an accuracy three orders of magnitude higher for the inverse analysis of unknown loadings. (2) With a training time five orders of magnitude faster, the DFEM is validated to be five orders of magnitude more accurate in determining unknown material parameters. Furthermore, two cases validate DFEM as effective for three-dimensional engineering structures: (1) A damaged cantilever beam characterized by twenty heterogeneous materials with forty unknown parameters is efficiently solved. (2) A tunnel lining ring with sparse noisy data under unknown heterogeneous boundary loadings is successfully analyzed. These problems are solved in seconds, corroborating DFEM's potential for engineering applications. Additionally, the DFEM framework can be generalized to a Physics-Encoded Numerical Network (PENN) for further development and exploration.