A Stochastic--Geometric Theory of Scaling Laws in Grokking

Delayed generalization (\ie~grokking) refers to the phenomenon in which a neural network fits its training data early in training but only begins to generalize after a prolonged delay, often through an abrupt transition. Despite extensive empirical study, its underlying mechanism remains poorly understood. In this work, we first theoretically characterize a shell--core topological configuration of the reachable solution space induced by Adam's optimization dynamics with weight-shrinkage regularization, supported by empirical evidence. This optimization-induced topological configuration gives rise to grokking. In model's parameter space, random initialization solutions concentrate on a thin outer spherical shell, enclosing another spherical shell of memorization solutions, which in turn contains a core corresponding to the generalization solutions. Leveraging stopping-time theory, we then analyze the geometry of this topological configuration and the solution transition time at which optimization trajectories escape the memorization manifold and first reach the boundary of the generalization manifold. Our theoretical analysis derives grokking scaling laws for the learning rate, batch size, and $\ell_2$ regularization coefficient, which are further validated through experiments and shown to recover results from prior literature.

Learning long term climate-resilient transport adaptation pathways under direct and indirect flood impacts using reinforcement learning

Climate change is expected to intensify rainfall and other hazards, increasing disruptions in urban transportation systems. Designing effective adaptation strategies is challenging due to the long-term, sequential nature of infrastructure investments, deep uncertainty, and complex cross-sector interactions. We propose a generic decision-support framework that couples an integrated assessment model (IAM) with reinforcement learning (RL) to learn adaptive, multi-decade investment pathways under uncertainty. The framework combines long-term climate projections (e.g., IPCC scenario pathways) with models that map projected extreme-weather drivers (e.g. rain) into hazard likelihoods (e.g. flooding), propagate hazards into urban infrastructure impacts (e.g. transport disruption), and value direct and indirect consequences for service performance and societal costs. Embedded in a reinforcement-learning loop, it learns adaptive climate adaptation policies that trade off investment and maintenance expenditures against avoided impacts. In collaboration with Copenhagen Municipality, we demonstrate the approach on pluvial flooding in the inner city for the horizon of 2024 to 2100. The learned strategies yield coordinated spatial-temporal pathways and improved robustness relative to conventional optimization baselines, namely inaction and random action, illustrating the framework's transferability to other hazards and cities.

Using Reinforcement Learning to Integrate Subjective Wellbeing into Climate Adaptation Decision Making

Subjective wellbeing is a fundamental aspect of human life, influencing life expectancy and economic productivity, among others. Mobility plays a critical role in maintaining wellbeing, yet the increasing frequency and intensity of both nuisance and high-impact floods due to climate change are expected to significantly disrupt access to activities and destinations, thereby affecting overall wellbeing. Addressing climate adaptation presents a complex challenge for policymakers, who must select and implement policies from a broad set of options with varying effects while managing resource constraints and uncertain climate projections. In this work, we propose a multi-modular framework that uses reinforcement learning as a decision-support tool for climate adaptation in Copenhagen, Denmark. Our framework integrates four interconnected components: long-term rainfall projections, flood modeling, transport accessibility, and wellbeing modeling. This approach enables decision-makers to identify spatial and temporal policy interventions that help sustain or enhance subjective wellbeing over time. By modeling climate adaptation as an open-ended system, our framework provides a structured framework for exploring and evaluating adaptation policy pathways. In doing so, it supports policymakers to make informed decisions that maximize wellbeing in the long run.

Climate Adaptation with Reinforcement Learning: Experiments with Flooding and Transportation in Copenhagen

Due to climate change the frequency and intensity of extreme rainfall events, which contribute to urban flooding, are expected to increase in many places. These floods can damage transport infrastructure and disrupt mobility, highlighting the need for cities to adapt to escalating risks. Reinforcement learning (RL) serves as a powerful tool for uncovering optimal adaptation strategies, determining how and where to deploy adaptation measures effectively, even under significant uncertainty. In this study, we leverage RL to identify the most effective timing and locations for implementing measures, aiming to reduce both direct and indirect impacts of flooding. Our framework integrates climate change projections of future rainfall events and floods, models city-wide motorized trips, and quantifies direct and indirect impacts on infrastructure and mobility. Preliminary results suggest that our RL-based approach can significantly enhance decision-making by prioritizing interventions in specific urban areas and identifying the optimal periods for their implementation.