Integrating Physical Models with Deep Learning for Neuroimaging Reconstruction and Materials Engineering

This thesis develops two novel statistical and machine learning methodologies that integrate Functional Data Analysis (FDA) and Physics-Informed Neural Networks (PINNs) for applications in neuroimaging and materials engineering. The unifying theme is the use of physical models to guide data-driven learning, producing accurate solutions in noisy contexts. The first contribution is called Functional Data Analysis Physics-Informed Neural Network (3D-PINNS) and is designed for reconstructing brain activity signals from functional Magnetic Resonance Imaging (fMRI) data. Extending Spatial Regression with Partial Differential Equation Regularization (SR-PDE), the method replaces traditional finite element basis functions with a neural network architecture, leveraging its high approximation power while retaining FDA techniques for smoothing parameter selection and uncertainty quantification. Extensive simulation studies on a spherical domain and a real brain surface demonstrate that 3D-PINNS outperform standard approaches such as Heat Kernel smoothing (HK), kriging, and standard neural networks. An application to Human Connectome Project (HCP) data illustrates the model’s capacity to recover smooth brain activity signals from noisy task-based fMRI measurements. The second contribution addresses nonlinear elasticity in materials engineering through an extension of the Deep Energy Method (DEM). Standard DEM approximates equilibrium states of elastic materials via energy minimization but does not directly incorporate empirical data. To overcome this limitation, the thesis introduces dataDEM, a hybrid PINN framework that integrates observational data with governing elasticity Partial Differential Equations (PDEs) into the cost function. Applied to a nonlinear elastic cantilever beam, dataDEM demonstrates the ability to reconcile noisy measurements with physical constraints, recovering accurate displacement fields and stress distributions. By bridging FDA and deep learning, this thesis establishes new methodologies for reconstructing functional data in neuroscience and for modeling the nonlinear behavior of elastic materials. The proposed frameworks highlight the value of embedding physical laws into machine learning models, enabling reliable solutions across disciplines.