Zahra Sharif Khodaei

Jinshuai Bai, Haolin Li, Zahra Sharif Khodaei et al.

Neural operator learning accelerates PDE solution by approximating operators as mappings between continuous function spaces. Yet in many engineering settings, varying geometry induces discrete structural changes, including topological changes, abrupt changes in boundary conditions or boundary types, and changes in the computational domain, which break the smooth-variation premise. Here we introduce Discrete Solution Operator Learning (DiSOL), a complementary paradigm that learns discrete solution procedures rather than continuous function-space operators. DiSOL factorizes the solver into learnable stages that mirror classical discretizations: local contribution encoding, multiscale assembly, and implicit solution reconstruction on an embedded grid, thereby preserving procedure-level consistency while adapting to geometry-dependent discrete structures. Across geometry-dependent Poisson, advection-diffusion, linear elasticity, as well as spatiotemporal heat conduction problems, DiSOL produces stable and accurate predictions under both in-distribution and strongly out-of-distribution geometries, including discontinuous boundaries and topological changes. These results highlight the need for procedural operator representations in geometry-dominated problems and position discrete solution operator learning as a distinct, complementary direction in scientific machine learning.

Dong Xiao, Zahra Sharif Khodaei, M. H. Aliabadi

This paper presents the first systematic application of a material homotopy continuation framework for efficient, automated computation of dispersion curves in viscoelastic waveguides of arbitrary cross-section. A material homotopy continuously maps the original lossy problem to an auxiliary lossless one via an attenuation parameter s in [0,1], addressing the core challenges of the non-Hermitian eigenvalue problem. Grounded in analytic perturbation theory, the method guarantees branch identity continuity--a one-to-one correspondence between solutions at s=0 and s=1--provided the real-parameter path does not cross any exceptional points. Under a Type I exceptional point topology, physical mode labels established at the elastic stage remain valid at the viscoelastic stage without post-processing, yielding the characteristic real-part veering with imaginary-part crossing. The decoupling strategy performs reliable mode tracking in the Hermitian regime via adaptive wavenumber refinement, then propagates a sparse set of key solutions to the target viscoelastic state through predictor-corrector homotopy continuation. Numerical examples across symmetric and unsymmetric laminates validate the framework's robustness and efficiency, with the majority of cases verified at a loss factor of approximately 0.003 and a single symmetric laminate providing additional support at 0.02. For a challenging unsymmetric laminate at a loss factor of 0.05, the method still produces numerically accurate solutions; two complementary diagnostic signatures--an extremely sharp imaginary-part crossing and a discernible discrepancy between spectral group velocity and energy flux velocity--warn of potential label mismatch and guide further analysis.

Haolin Li, Yuyang Miao, Zahra Sharif Khodaei et al.

PINN models have demonstrated capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite domain, which conflicts with the finite boundaries typical of most solid structures; and b) the solution space utilised by PINN is Euclidean, which is inadequate for addressing the complex geometries often present in solid structures. This work presents a PINN architecture for general solid mechanics problems, referred to as the Finite-PINN model. The model is designed to effectively tackle two key challenges, while retaining as much of the original PINN framework as possible. To this end, the Finite-PINN incorporates finite geometric encoding into the neural network inputs, thereby transforming the solution space from a conventional Euclidean space into a hybrid Euclidean-topological space. The model is comprehensively trained using both strong-form and weak-form loss formulations, enabling its application to a wide range of forward and inverse problems in solid mechanics. For forward problems, the Finite-PINN model efficiently approximates solutions to solid mechanics problems when the geometric information of a given structure has been preprocessed. For inverse problems, it effectively reconstructs full-field solutions from very sparse observations by embedding both physical laws and geometric information within its architecture.