Adam J. Thorpe

Adam J. Thorpe, Stepan Tretiakov, Cheng-Hsi Hsiao et al.

World models for embodied AI must be physically viable: constructed to answer intervention queries by representing the physical structure governing action outcomes, rather than merely predicting future observations. Existing observation-predictive world models can produce visually plausible but physically wrong rollouts. This failure is structural; distinct physical systems can look identical yet diverge under intervention. We expose this problem with controlled benchmarks that fix the visible scene while varying latent physics. We show that such models may recommend infeasible actions, mispredict interaction outcomes, or certify unsafe behavior. We argue that embodied AI requires world models that identify the simplest physical abstraction sufficient to answer an intervention query. Such a model comprises modular components, including environment representation, latent state and parameter estimation, action specification, interventional dynamics, and query-level response. An autonomous orchestrator should identify the relevant abstraction and compose compatible learned and structured components per query. When closed-form physics is unavailable, uncertain, or costly, the transition model may be analytic, simulated, learned, or hybrid, but it must preserve the structure that determines interventional outcomes. This decomposition makes the model interpretable, its components verifiable, and its outputs auditable against the query. It also provides a design principle for new world models and a feasibility test for existing ones: the right abstraction is not the most detailed model of the world, but the simplest model that preserves the distinctions relevant to the query. We demonstrate this approach on queries that existing systems fail to answer correctly, and outline how an orchestrator can dynamically assemble and adapt physically viable models for planning, control, and verification.

Tyler Ingebrand, Adam J. Thorpe, Somdatta Goswami et al.

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.

William Ward, Sarah Etter, Jesse Quattrociocchi et al.

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. This yields constant-time coefficient estimation without gradient-based inner-loop updates, enabling adaptation from only a few seconds of data. We evaluate our approach on a Van 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.

Adam J. Thorpe, Cyrus Neary, Franck Djeumou et al.

Data-driven control algorithms use observations of system dynamics to construct an implicit model for the purpose of control. However, in practice, data-driven techniques often require excessive sample sizes, which may be infeasible in real-world scenarios where only limited observations of the system are available. Furthermore, purely data-driven methods often neglect useful a priori knowledge, such as approximate models of the system dynamics. We present a method to incorporate such prior knowledge into data-driven control algorithms using kernel embeddings, a nonparametric machine learning technique based in the theory of reproducing kernel Hilbert spaces. Our proposed approach incorporates prior knowledge of the system dynamics as a bias term in the kernel learning problem. We formulate the biased learning problem as a least-squares problem with a regularization term that is informed by the dynamics, that has an efficiently computable, closed-form solution. Through numerical experiments, we empirically demonstrate the improved sample efficiency and out-of-sample generalization of our approach over a purely data-driven baseline. We demonstrate an application of our method to control through a target tracking problem with nonholonomic dynamics, and on spring-mass-damper and F-16 aircraft state prediction tasks.

Adam J. Thorpe, Meeko M. K. Oishi

We present SOCKS, a data-driven stochastic optimal control toolbox based in kernel methods. SOCKS is a collection of data-driven algorithms that compute approximate solutions to stochastic optimal control problems with arbitrary cost and constraint functions, including stochastic reachability, which seeks to determine the likelihood that a system will reach a desired target set while respecting a set of pre-defined safety constraints. Our approach relies upon a class of machine learning algorithms based in kernel methods, a nonparametric technique which can be used to represent probability distributions in a high-dimensional space of functions known as a reproducing kernel Hilbert space. As a nonparametric technique, kernel methods are inherently data-driven, meaning that they do not place prior assumptions on the system dynamics or the structure of the uncertainty. This makes the toolbox amenable to a wide variety of systems, including those with nonlinear dynamics, black-box elements, and poorly characterized stochastic disturbances. We present the main features of SOCKS and demonstrate its capabilities on several benchmarks.

Adam J. Thorpe, Stepan Tretiakov, Dibakar Roy Sarkar et al.

Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B${}^{-1}$, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness.

Tyler Ingebrand, Adam J. Thorpe, Ufuk Topcu

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.

Su Ann Low, Quentin Rommel, Kevin S. Miller et al.

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.

Kevin S. Miller, Adam J. Thorpe, Ufuk Topcu

We present an active learning algorithm for learning dynamics that leverages side information by explicitly incorporating prior domain knowledge into the sampling process. Our proposed algorithm guides the exploration toward regions that demonstrate high empirical discrepancy between the observed data and an imperfect prior model of the dynamics derived from side information. Through numerical experiments, we demonstrate that this strategy explores regions of high discrepancy and accelerates learning while simultaneously reducing model uncertainty. We rigorously prove that our active learning algorithm yields a consistent estimate of the underlying dynamics by providing an explicit rate of convergence for the maximum predictive variance. We demonstrate the efficacy of our approach on an under-actuated pendulum system and on the half-cheetah MuJoCo environment.

Adam J. Thorpe, Thomas Lew, Meeko M. K. Oishi et al.

We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finite subset of stochastic open-loop control trajectories, we relax the original problem to a linear program over the control parameters that can be efficiently solved using standard convex optimization techniques. We demonstrate our proposed approach in simulation on a system with nonlinear non-Markovian dynamics navigating in a cluttered environment.