Deep Learning Accelerated Numerical Methods for PDEs in Engineering Simulations

Thesis Abstract

This study investigates the use of deep learning surrogates to accelerate numerical solutions of partial differential equations (PDEs) in complex engineering simulations, addressing the computational bottleneck inherent in high-fidelity, physics-based solvers. The problem arises from the prohibitive cost of repeated PDE solves in multi-physics contexts, real-time control, and optimization loops, where conventional finite element, finite volume, or spectral methods incur substantial wall-clock times and memory demands. The aim is to develop a robust, data-informed framework that integrates deep learning with traditional numerical methods to reduce solve times while preserving accuracy and stability across diverse engineering regimes. Specific objectives are (1) to design neural network surrogates trained on high-fidelity solver data for steady and transient PDEs common in structural, fluid, and thermal analyses; (2) to formulate physics-informed loss functions and operator learning architectures that generalize across mesh resolutions, boundary conditions, and material heterogeneities; (3) to establish a hybrid solver workflow that couples learned components with classical solvers to maintain rigorous error bounds; (4) to evaluate performance on benchmark problems including incompressible Navier–Stokes, elasticity, and convection–diffusion systems; and (5) to compare the proposed approach against state-of-the-art reduced-order models (ROMs) and multilevel Monte Carlo methods in terms of accuracy, speedup, and stability under parameter perturbations. The methodology adopts a mixed-methods research design combining empirical computational experiments with theoretical analysis. The population comprises PDE solution instances generated from high-fidelity simulations using commercial and open-source solvers (e.g., FEniCS for FEM, OpenFOAM for CFD) across three engineering domains aerodynamics, structural mechanics, and heat transfer. A stratified sampling scheme yields a training set of 50,000 PDE solves for Navier–Stokes and 30,000 for linear and nonlinear elasticity problems, with varying Reynolds numbers, load magnitudes, and mesh densities. Data collection instruments include solver dashboards, numerical residuals, and solution fields stored in HDF5 format, alongside metadata describing boundary conditions and material properties. Training employs physics-constrained neural networks, operator-learning architectures such as DeepONet and Fourier Neural Operators (FNO), and graph neural networks (GNNs) to handle unstructured meshes. Validation uses a held-out test set of 20,000 Navier–Stokes solves and 15,000 elasticity solves, including unseen geometries to assess generalization. The analysis combines regression-based error quantification, relative L2 and energy-norm errors, and time-to-solution comparisons. Statistical techniques include paired t-tests to assess speedups, Bland–Altman plots for error agreement, and ANOVA to evaluate influence of mesh size and boundary-condition variation on surrogate accuracy. Theoretical underpinnings draw on operator theory and Bayesian inference for uncertainty quantification, with attention to stability via contractive mapping properties and energy dissipation principles. The study also incorporates an ethical framework for reproducibility, data sharing, and software provenance. The analytical framework comprises a hybrid solver architecture a learned operator approximates the action of the PDE inverse on coarse representations, while a classical solver corrects residuals to enforce compatibility with conservation laws. Model specification includes a dual-branch architecture (i) a global surrogate predicting coarse-scale fields, and (ii) a local refinement module that interpolates fine-scale corrections. Regularization employs physics-informed losses, including divergence-free constraints for incompressible flow and energy-based penalties for elastic deformations. Uncertainty is addressed through ensemble methods and probabilistic neural networks to provide confidence intervals around predictions. Evaluation metrics extend beyond accuracy to include stability under perturbations, scalability with mesh refinement, and robustness to missing boundary data. Expected findings indicate substantial speedups (2–10×) with acceptable accuracy losses (less than 3% L2 relative error on average) and improved resilience to mesh-coarsening compared with traditional ROMs. The study anticipates that operator-learning surrogates will generalize across related PDE families when guided by physics-informed constraints and sufficient diversity in training data. The contribution to knowledge is threefold (1) a validated hybrid physics-informed deep learning framework for accelerating PDE solves in engineering simulations, (2) a comparative performance analysis against ROMs and multilevel methods highlighting regimes where learning-based surrogates outperform classical approaches, and (3) practical guidelines for integrating learned operators into production solvers with uncertainty quantification and error-bounding mechanisms. The main conclusion is that carefully designed deep learning surrogates can dramatically reduce computational cost while preserving essential physical fidelity, provided that surrogate models are anchored by conservation laws and validated across representative engineering scenarios. Recommendations include extending the framework to coupled multiphysics problems, incorporating active learning to refine models in regions of high residuals, and developing standardized benchmarks to facilitate cross-domain adoption.