This challenge necessitates new mathematical and algorithmic tools that enable adaptation and zero-shot transfer.

This challenge necessitates new mathematical and algorithmic tools that enable adaptation and zero-shot transfer.

Function encoders are a recent technique that learn neural network basis functions to form compact, adaptive representations of Hilbert spaces of functions. We show that function encoders provide a principled connection to feature learning and kernel methods by defining a kernel through an inner product of the learned feature map. This kernel-theoretic perspective explains their ability to scale independently of dataset size while adapting to the intrinsic structure of data, and it enables kernel-style analysis of neural models. Building on this foundation, we develop two training algorithms that learn compact bases: a progressive training approach that constructively grows bases, and a train-then-prune approach that offers a computationally efficient alternative after training. Both approaches use principles from PCA to reveal the intrinsic dimension of the learned space. In parallel, we derive finite-sample generalization bounds using Rademacher complexity and PAC-Bayes techniques, providing inference time guarantees. We validate our approach on a polynomial benchmark with a known intrinsic dimension, and on nonlinear dynamical systems including a Van der Pol oscillator and a two-body orbital model, demonstrating that the same accuracy can be achieved with substantially fewer basis functions. This work suggests a path toward neural predictors with kernel-level guarantees, enabling adaptable models that are both efficient and principled at scale.

Abstract--Autonomous robots must go from zero prior knowledge to safe control within seconds to operate in unstructured environments. Abrupt terrain changes, such as a sudden transition to ice, create dynamics shifts that can destabilize planners unless the model adapts in real-time. We present a method for online adaptation that combines function encoders with recursive least squares, treating the function encoder coefficients as latent states updated from streaming odometry. We evaluate our approach on a V an der Pol system to highlight algorithmic behavior, in a Unity simulator for high-fidelity off-road navigation, and on a Clearpath Jackal robot, including on a challenging terrain at a local ice rink. Across these settings, our method improves model accuracy and downstream planning, reducing collisions compared to static and meta-learning baselines. High-speed ground vehicles require dynamics models that evolve as quickly as the terrain itself. When operating near the limits of controllability, even modest prediction errors in ground terrain interaction can lead to instability, skidding, or rollover. This problem is particularly apparent in off-road navigation: transitions such as pavement to loose gravel can change friction properties within seconds, while mixed terrain features introduce variation in the terrain properties that are difficult to accurately predict. Planning frameworks such as Model Predictive Path Integral Control (MPPI) [27] rely on an accurate model of the system dynamics to predict rollouts and select optimal control actions in real-time.

--Autonomous mobile robots operating in remote, unstructured environments must adapt to new, unpredictable terrains that can change rapidly during operation. In such scenarios, a critical challenge becomes estimating the robot's dynamics on changing terrain in order to enable reliable, accurate navigation and planning. We present a novel online adaptation approach for terrain-aware dynamics modeling and planning using function encoders. By learning a set of neural network basis functions that span the robot dynamics on diverse terrains, we enable rapid online adaptation to new, unseen terrains and environments as a simple least-squares calculation. We demonstrate our approach for terrain adaptation in a Unity-based robotics simulator and show that the downstream controller has better empirical performance due to higher accuracy of the learned model. This leads to fewer collisions with obstacles while navigating in cluttered environments as compared to a neural ODE baseline. Rapid adaptation to unknown environments and terrain is critical for autonomous mobile robots. In off-road navigation, unpredictable terrain features such as rocky paths, forest floors, and wet fields can cause skidding, tripping, or immobilization, jeopardizing the robot's ability to reach its objective. Autonomous ground vehicles must therefore dynamically adjust their behavior to terrain-specific conditions. This adaptation is challenging because terrain variations directly alter system dynamics. For example, tire response to acceleration depends on surface friction.

Variations in hidden parameters, such as a robot's mass distribution or friction, pose safety risks during execution. We develop a runtime shielding mechanism for reinforcement learning, building on the formalism of constrained hidden-parameter Markov decision processes. Function encoders enable real-time inference of hidden parameters from observations, allowing the shield and the underlying policy to adapt online. The shield constrains the action space by forecasting future safety risks (such as obstacle proximity) and accounts for uncertainty via conformal prediction. We prove that the proposed mechanism satisfies probabilistic safety guarantees and yields optimal policies among the set of safety-compliant policies. Experiments across diverse environments with varying hidden parameters show that our method significantly reduces safety violations and achieves strong out-of-distribution generalization, while incurring minimal runtime overhead.

A central challenge in transfer learning is designing algorithms that can quickly adapt and generalize to new tasks without retraining. Yet, the conditions of when and how algorithms can effectively transfer to new tasks is poorly characterized. We introduce a geometric characterization of transfer in Hilbert spaces and define three types of inductive transfer: interpolation within the convex hull, extrapolation to the linear span, and extrapolation outside the span. We propose a method grounded in the theory of function encoders to achieve all three types of transfer. Specifically, we introduce a novel training scheme for function encoders using least-squares optimization, prove a universal approximation theorem for function encoders, and provide a comprehensive comparison with existing approaches such as transformers and meta-learning on four diverse benchmarks. Our experiments demonstrate that the function encoder outperforms state-of-the-art methods on four benchmark tasks and on all three types of transfer.

We present Basis-to-Basis (B2B) operator learning, a novel approach for learning operators on Hilbert spaces of functions based on the foundational ideas of function encoders. We decompose the task of learning operators into two parts: learning sets of basis functions for both the input and output spaces and learning a potentially nonlinear mapping between the coefficients of the basis functions. B2B operator learning circumvents many challenges of prior works, such as requiring data to be at fixed locations, by leveraging classic techniques such as least squares to compute the coefficients. It is especially potent for linear operators, where we compute a mapping between bases as a single matrix transformation with a closed-form solution. Furthermore, with minimal modifications and using the deep theoretical connections between function encoders and functional analysis, we derive operator learning algorithms that are directly analogous to eigen-decomposition and singular value decomposition. We empirically validate B2B operator learning on seven benchmark operator learning tasks and show that it demonstrates a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks.

Autonomous systems often encounter environments and scenarios beyond the scope of their training data, which underscores a critical challenge: the need to generalize and adapt to unseen scenarios in real time. This challenge necessitates new mathematical and algorithmic tools that enable adaptation and zero-shot transfer. To this end, we leverage the theory of function encoders, which enables zero-shot transfer by combining the flexibility of neural networks with the mathematical principles of Hilbert spaces. Using this theory, we first present a method for learning a space of dynamics spanned by a set of neural ODE basis functions. After training, the proposed approach can rapidly identify dynamics in the learned space using an efficient inner product calculation. Critically, this calculation requires no gradient calculations or retraining during the online phase. This method enables zero-shot transfer for autonomous systems at runtime and opens the door for a new class of adaptable control algorithms. We demonstrate state-of-the-art system modeling accuracy for two MuJoCo robot environments and show that the learned models can be used for more efficient MPC control of a quadrotor.

Although reinforcement learning (RL) can solve many challenging sequential decision making problems, achieving zero-shot transfer across related tasks remains a challenge. The difficulty lies in finding a good representation for the current task so that the agent understands how it relates to previously seen tasks. To achieve zero-shot transfer, we introduce the function encoder, a representation learning algorithm which represents a function as a weighted combination of learned, non-linear basis functions. By using a function encoder to represent the reward function or the transition function, the agent has information on how the current task relates to previously seen tasks via a coherent vector representation. Thus, the agent is able to achieve transfer between related tasks at run time with no additional training. We demonstrate state-of-the-art data efficiency, asymptotic performance, and training stability in three RL fields by augmenting basic RL algorithms with a function encoder task representation.