Data-driven spectral methods for sparse PDE solvers: design, implement, evaluate

Thesis Abstract

The increasing demand for efficient and scalable solvers for sparse partial differential equations (PDEs) in high-dimensional and time-dependent problems motivates the development of data-driven spectral methods that can adaptively capture dominant modes while preserving numerical stability. This study addresses the gap between classical spectral techniques, which rely on fixed bases, and contemporary data-driven approaches, which often neglect the structure of sparse discretizations. The aim is to design, implement, and evaluate a data-driven spectral framework that leverages learned basis representations and operator compression to accelerate sparse PDE solvers without sacrificing accuracy or robustness. The specific objectives are to (1) formulate a spectral representation that integrates learned dictionaries with conventional Fourier and Chebyshev bases, (2) develop an algorithm for adaptive basis selection and operator compression that preserves sparsity patterns, (3) implement the framework within a modular solver architecture for linear and nonlinear PDEs, and (4) evaluate performance across benchmark problems in fluid dynamics and elasticity. The study adopts a design-based, empirical research approach grounded in numerical analysis and machine learning theory. The population consists of sparse discretizations of canonical PDEs the 2D and 3D Poisson equation, the incompressible Navier–Stokes equations in a periodic box, and nonlinear reaction–diffusion systems. A sample of 15 representative problem instances is generated by varying mesh resolutions (from 64×64×64 to 256×256×256 in 3D) and time horizons, yielding approximately 120 solver runs. Data collection instruments include (i) a high-fidelity baseline solver (FFT-based spectral method and finite element method as applicable), (ii) the proposed data-driven spectral solver with learned dictionaries trained on Jacobian and Hessian samples, and (iii) performance metrics recorded during solver iterations. Training data for the dictionaries are obtained from precomputed operator evaluations and modal coefficients across initial condition ensembles, totaling about 2,000 operator evaluations. Data analysis proceeds in three stages. First, regression and model-selection techniques are employed to identify the optimal mixed basis configuration, comparing fixed bases (Fourier, Chebyshev) against the learned dictionary approach using cross-validated sparse regression (LASSO) and elastic net. Second, spectral accuracy and operator compression quality are quantified via norm-based error metrics (L2 and energy-norm errors) and compression ratios, with statistical significance assessed through paired t-tests across problem classes. Third, solver performance is analyzed through convergence rate examination and total wall-clock time, employing ANOVA to determine effect sizes of basis choice, mesh resolution, and problem class on computational efficiency. Additionally, stability analysis is conducted via spectrum plots and residual norms to detect potential aliasing or numerical instabilities. Key expected findings include (i) improved convergence rates for linear and mildly nonlinear PDEs when incorporating learned spectral components, (ii) substantial reductions in operator storage and multiplication costs due to targeted sparsity-preserving compression, and (iii) robust performance across heterogeneous problem settings with limited degradation in accuracy relative to baseline methods. The framework is anticipated to outperform traditional fixed-basis solvers on high-dimensional problems with anisotropic features, owing to adaptive representation capacity and better exploitation of sparsity patterns. The study contributes to knowledge by bridging data-driven representation learning with established spectral discretization techniques for sparse PDEs, providing a repeatable methodology for integrating learned bases into solver pipelines, and delivering empirical evidence on accuracy–efficiency trade-offs. Theoretical underpinnings draw on spectral approximation theory, compressed sensing, and perturbation analysis for sparse operators, with the practical design guided by the theoretical stability criteria akin to the Lax–Rieder framework for numerical solvers. The main conclusion is that data-driven spectral methods can substantially enhance solver efficiency for sparse PDEs while maintaining mathematical fidelity, given careful design of basis adaptation and operator compression. Recommendations include extending the framework to heterogeneous hardware architectures (GPUs and distributed memory), exploring unsupervised dictionary adaptation during time integration, and applying the approach to multi-physics PDE systems to assess transferability of learned representations.