Scalar field features such as extrema, contours, and saddle points are essential for applications in environmental monitoring, search and rescue, and resource exploration. Traditional navigation methods often rely on predefined trajectories, leading to inefficient and resource-intensive mapping. This paper introduces a new adaptive navigation framework that leverages learning techniques to enhance exploration efficiency and effectiveness in scalar fields, even under noisy data and obstacles. The framework employs Partial Differential Equations to model scalar fields and a Gaussian Process Regressor to estimate the fields and their gradients, enabling real-time path adjustments and obstacle avoidance. We provide a theoretical foundation for the approach and address several limitations found in existing methods. The effectiveness of our framework is demonstrated through simulation benchmarks and field experiments with an Autonomous Surface Vehicle, showing improved efficiency and adaptability compared to traditional methods and offering a robust solution for real-time environmental monitoring.

Predicting the near-term behavior of a reactive agent is crucial in many robotic scenarios, yet remains challenging when observations of that agent are sparse or intermittent. Vision-Language Models (VLMs) offer a promising avenue by integrating textual domain knowledge with visual cues, but their one-shot predictions often miss important edge cases and unusual maneuvers. Our key insight is that iterative, counterfactual exploration–where a dedicated module probes each proposed behavior hypothesis, explicitly represented as a plausible trajectory, for overlooked possibilities–can significantly enhance VLM-based behavioral forecasting. We present TRACE (Tree-of-thought Reasoning And Counterfactual Exploration), an inference framework that couples tree-of-thought generation with domain-aware feedback to refine behavior hypotheses over multiple rounds. Concretely, a VLM first proposes candidate trajectories for the agent; a counterfactual critic then suggests edge-case variations consistent with partial observations, prompting the VLM to expand or adjust its hypotheses in the next iteration. This creates a self-improving cycle where the VLM progressively internalizes edge cases from previous rounds, systematically uncovering not only typical behaviors but also rare or borderline maneuvers, ultimately yielding more robust trajectory predictions from minimal sensor data. We validate TRACE on both ground-vehicle simulations and real-world marine autonomous surface vehicles. Experimental results show that our method consistently outperforms standard VLM-driven and purely model-based baselines, capturing a broader range of feasible agent behaviors despite sparse sensing. Evaluation videos and code are available at trace-robotics.github.io.