Alzheimer's disease (AD) is a complex, multifactorial neurodegenerative disorder with substantial heterogeneity in progression and treatment response. Despite recent therapeutic advances, predictive models capable of accurately forecasting individualized disease trajectories remain limited. Here, we present a machine learning-based operator learning framework for personalized modeling of AD progression, integrating longitudinal multimodal imaging, biomarker, and clinical data. Unlike conventional models with prespecified dynamics, our approach directly learns patient-specific disease operators governing the spatiotemporal evolution of amyloid, tau, and neurodegeneration biomarkers. Using Laplacian eigenfunction bases, we construct geometry-aware neural operators capable of capturing complex brain dynamics. Embedded within a digital twin paradigm, the framework enables individualized predictions, simulation of therapeutic interventions, and in silico clinical trials. Applied to AD clinical data, our method achieves high prediction accuracy exceeding 90% across multiple biomarkers, substantially outperforming existing approaches. This work offers a scalable, interpretable platform for precision modeling and personalized therapeutic optimization in neurodegenerative diseases.

In reinforcement learning, universal successor features (SFs) are a way to provide zero-shot adaptation to new tasks at test time: they provide optimal policies for all downstream reward functions lying in the linear span of a set of base features. But it is unclear what constitutes a good set of base features, that could be useful for a wide set of downstream tasks beyond their linear span. Laplacian eigenfunctions (the eigenfunctions of $\Delta+\Delta^\ast$ with $\Delta$ the Laplacian operator of some reference policy and $\Delta^\ast$ that of the time-reversed dynamics) have been argued to play a role, and offer good empirical performance. Here, for the first time, we identify the optimal base features based on an objective criterion of downstream performance, in a non-tautological way without assuming the downstream tasks are linear in the features. We do this for three generic classes of downstream tasks: reaching a random goal state, dense random Gaussian rewards, and random ``scattered'' sparse rewards. The features yielding optimal expected downstream performance turn out to be the \emph{same} for these three task families. They do not coincide with Laplacian eigenfunctions in general, though they can be expressed from $\Delta$: in the simplest case (deterministic environment and decay factor $\gamma$ close to $1$), they are the eigenfunctions of $\Delta^{-1}+(\Delta^{-1})^\ast$. We obtain these results under an assumption of large behavior cloning regularization with respect to a reference policy, a setting often used for offline RL. Along the way, we get new insights into KL-regularized\option{natural} policy gradient, and into the lack of SF information in the norm of Bellman gaps.

We investigate the problem of automatically constructing efficient representations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigenfunctions of the Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying representation and policies are simultaneously learned.

We investigate the problem of automatically constructing efficient representations or basis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, two novel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigenfunctions of the Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying representation and policies are simultaneously learned.

We investigate the problem of automatically constructing efficient representations orbasis functions for approximating value functions based on analyzing the structure and topology of the state space. In particular, twonovel approaches to value function approximation are explored based on automatically constructing basis functions on state spaces that can be represented as graphs or manifolds: one approach uses the eigenfunctions ofthe Laplacian, in effect performing a global Fourier analysis on the graph; the second approach is based on diffusion wavelets, which generalize classical wavelets to graphs using multiscale dilations induced by powers of a diffusion operator or random walk on the graph. Together, these approaches form the foundation of a new generation of methods for solving large Markov decision processes, in which the underlying representation andpolicies are simultaneously learned.