Applied Math Seminar- Structure-Aware Scientific Machine Learning for Personalized Digital Twins: From Personalized Biomedical Inference to Reliable Neural Network Solvers for Nonlinear PDEs

Title: Structure-Aware Scientific Machine Learning for Personalized Digital Twins: From Personalized Biomedical Inference to Reliable Neural Network Solvers for Nonlinear PDEs

 

Abstract: Understanding complex dynamical systems from sparse, heterogeneous data requires models that learn effectively while respecting the mathematical structures governing their evolution. In this talk, I present a structure-informed scientific machine learning research program spanning personalized health inference and reliable neural solvers for nonlinear PDEs.

 

First, I introduce a personalized graph-PDE framework for modeling the spatiotemporal progression of Alzheimer’s disease from longitudinal neuroimaging data such as MRI, PET, and resting-state fMRI. The model represents 68 brain regions as nodes in a patient-specific functional-connectivity graph and uses a low-rank representation to capture individual deviations from population-level connectivity. It couples amyloid-β, tau, neurodegeneration, and cognitive decline through local interactions and network-mediated propagation. To enable identifiable parameter inference from sparse longitudinal data in this nonlinear model, I develop a hierarchical inference strategy and homotopy regularization. When applied to 1,891 participants from the Alzheimer’s Disease Neuroimaging Initiative (ADNI), the model improves the prediction of future biomarker trajectories over homogeneous and standard clinical or neuroimaging benchmarks, with gains of up to 9.73 percentage points for individuals whose brain networks differ substantially from the population norm.

 

Second, scalable personalized mechanistic models require accurate forward simulation and parameter inference for nonlinear, multiscale PDEs. Using thermodynamically consistent phase-field systems as a testbed, I develop methods that improve both where neural solvers allocate computational effort and how they are optimized. Energy-dissipation-rate-guided adaptive sampling (EDRAS) targets regions with high local energy dissipation where interfacial evolution occurs, rather than refining solely where the PDE residual is large. For Allen–Cahn dynamics, EDRAS reduces relative mean-square error by up to a factor of six compared with residual-based refinement and is more likely to identify high-error regions. I also develop random projection neural networks (RPNNs), which fix randomly generated hidden features and determine output coefficients through deterministic nonlinear least squares, enabling second-order optimization. For the fourth-order Cahn–Hilliard system, a scaled auxiliary-variable formulation, separate random-feature spaces, adaptive sampling, and domain decomposition yield relative $L^2$ errors as low as $8.64x10^{−4}$, outperforming matched deep-PINN baselines by nearly three orders of magnitude.

 

Together, these works advance interpretable, personalized mechanistic digital twins underpinned by reliable scientific machine learning.