Implicit Tensor-Train Cross Integration of High-Dimensional Nonlinear PDEs via Fiber-Dependency Elimination

Tensor-Train Methods for 3D Linear Elasticity: Block and Global Operator Representations with Solver Performance Analysis

This work develops tensor-train (TT) formulations for solving large-scale three-dimensional linear elasticity problems discretized by isogeometric analysis. By exploiting the tensor-product structure of the basis functions and the low-rank structure of geometry-dependent coefficient fields, the stiffness operator, mass operator, force vector, and displacement solution are represented in TT format. Two solution strategies are investigated: a block-operator formulation, in which the coupled elasticity operator is stored as separated TT blocks, and a single-operator formulation, in which the full coupled system is stored as one monolithic TT operator. A matrix-free three-field TT conjugate-gradient solver is introduced for the block formulation, while AMEn is used for the single-operator formulation. Numerical examples demonstrate substantial compression of both operators and solutions compared with conventional sparse full-grid representations, showing that TT-based formulations provide an efficient and scalable approach for large-scale three-dimensional elasticity simulations.

Alternating Iterative Methods for Solving the Inverse Eigenvalue Problems on the Manifold of Tensors with Fixed TT-Ranks

Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers

Physics-informed neural networks and finite element methods provide two different paradigms for the numerical approximation of partial differential equations: the former are commonly trained by minimizing pointwise strong residuals, whereas the latter are naturally built from weak variational formulations and the finite-dimensional systems obtained after discretization. In this work, we introduce a common framework based on the discretization of functional Gauss--Newton problems by finite families of linear measurements. We show that, through an appropriate duality pairing, the linear measurements can be represented by test functions. The resulting Gauss--Newton system is then precisely a Petrov--Galerkin discretization of the linearized functional problem. This perspective recovers pointwise collocation and natural-gradient constructions as particular cases, while making the choice of test functions an explicit algorithmic design choice. We specialize this framework to elliptic problems, where it naturally leads to weak residual formulations and to a hybrid finite element--neural construction acting on complementary approximation spaces. Numerical experiments support the proposed framework and demonstrate the effectiveness of weak Gauss--Newton formulations and hybrid finite element--neural approximations.

Dimensional hyperreduction of nonlinear finite element models via empirical cubature with manifold-adaptive weights

Nonlinear-manifold reduced-order models for parametrized finite element problems can achieve substantial compression both in the number of generalized (latent) coordinates and, through sampling-and-weighting hyperreduction, in the number of sampled elements/integration points. Yet current sampling-and-weighting approaches employ weights that remain fixed over the solution manifold. We contend that this restriction leaves hyperreduction potential untapped: allowing the weights to vary continuously and nonlinearly with the latent coordinates can further decrease the number of sampled spatial entities. To exploit this possibility, we propose the Manifold-Adaptive-Weight Empirical Cubature Method (MAW-ECM). Starting from a feasible fixed-weight ECM rule, a greedy pruning strategy removes sampled entities through convex quadratic weight-redistribution problems enforcing local conditions and positivity. The method is assessed on two nonlinear benchmarks: homogenization of a metamaterial unit cell exhibiting negative incremental stiffness, and a history-dependent continuum-damage problem. In both cases, the nonlinear manifold is constructed from an initial linear compression followed by an input-informed identification of the latent coordinates as general linear combinations of the retained modal amplitudes, incorporating graph information when relevant to seek the intrinsic dimensionality of the solution manifold. We show that combining the nonlinear-manifold representation with MAW-ECM reduces the number of sampled integration points by more than two orders of magnitude relative to the corresponding standard linear reduced model. Relative to the fixed-weight manifold models alone, the adaptive weights eliminate approximately 80% of the remaining points in the homogenization benchmark and more than 97% in the damage benchmark, while essentially preserving their accuracy.

Residual-Driven Lifting Identification for Nonlinear-Manifold Reduced-Order Models of Parametrized Linear PDEs

We introduce a residual-driven procedure for training nonlinear-manifold reduced-order models for parametrized linear partial differential equations that, given prescribed latent and lifting spaces, identifies the nonlinear lifting without high-fidelity solution snapshots. The approximation is represented by a low-dimensional latent coordinate together with a nonlinear lifting into a richer reduced space. Rather than fitting the lifting to snapshot data, we determine it by minimizing a computable residual-based upper bound for the state error. For affinely parametrized operators, the resulting training objective admits an efficient offline--online decomposition, and the lifting update reduces to a sequence of low-dimensional weighted least-squares problems. Numerical evaluations on an advection--diffusion problem and a plane-strain elasticity benchmark show that the proposed approach substantially improves accuracy over linear subspaces. The resulting nonlinear models achieve accuracy comparable to snapshot-driven nonlinear-manifold training while avoiding high-fidelity snapshots in the lifting-identification stage.

On the Removal of Solver-Induced Dependencies in Momentum-Weighted Interpolation for Primal and Continuous-Adjoint Flow Solvers

Momentum-Weighted Interpolation (MWI) is a key component in pressure--velocity coupling schemes on collocated cell-centered finite-volume methods for both primal and continuous adjoint formulations. In many practical implementations, MWI relies on diagonal momentum coefficients that include contributions from under-relaxation and time discretization. As a result, both primal quantities of interest and adjoint sensitivities may exhibit a non-physical dependence on solver parameters such as relaxation factors and time-step size, and no well-defined limit is obtained as these parameters approach zero. In this work, building on previous developments in discrete-consistent MWI formulations, a simple correction is proposed that removes solver-induced contributions from the diagonal momentum coefficients in the pressure-driven term. The resulting formulation preserves the original discretization while eliminating artificial dependencies on relaxation and time-stepping parameters and is applied consistently to both primal and adjoint systems. To facilitate its application, the derivation is presented in a structured, recipe-like manner that can be readily followed and transferred to different finite volume-based solver configurations. The proposed modification is assessed for a two-dimensional laminar cylinder flow and a three-dimensional turbulent ship hull flow configuration. In both cases, the uncorrected formulation leads to significant variations in forces, wake-related quantities, and shape sensitivities when solver parameters are altered, despite all simulations being iterated to converged residual levels and stable integral quantities. In contrast, the corrected formulation yields consistent results across a wide range of relaxation factors and time-step sizes.