Physically Viable World Models: A Case for Query-Conditioned Embodied AI

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.

Neural Operators for Multi-Task Control and Adaptation

Neural operator methods have emerged as powerful tools for learning mappings between infinite-dimensional function spaces, yet their potential in optimal control remains largely unexplored. We focus on multi-task control problems, whose solution is a mapping from task description (e.g., cost or dynamics functions) to optimal control law (e.g., feedback policy). We approximate these solution operators using a permutation-invariant neural operator architecture. Across a range of parametric optimal control environments and a locomotion benchmark, a single operator trained via behavioral cloning accurately approximates the solution operator and generalizes to unseen tasks, out-of-distribution settings, and varying amounts of task observations. We further show that the branch-trunk structure of our neural operator architecture enables efficient and flexible adaptation to new tasks. We develop structured adaptation strategies ranging from lightweight updates to full-network fine-tuning, achieving strong performance across different data and compute settings. Finally, we introduce meta-trained operator variants that optimize the initialization for few-shot adaptation. These methods enable rapid task adaptation with limited data and consistently outperform a popular meta-learning baseline. Together, our results demonstrate that neural operators provide a unified and efficient framework for multi-task control and adaptation.

Learning Generalizable Neural Operators for Inverse Problems

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.

SetONet: A Deep Set-based Operator Network for Solving PDEs with permutation invariant variable input sampling

Neural operators, particularly the Deep Operator Network (DeepONet), have shown promise in learning mappings between function spaces for solving differential equations. However, standard DeepONet requires input functions to be sampled at fixed locations, limiting its applicability in scenarios with variable sensor configurations, missing data, or irregular grids. We introduce the Set Operator Network (SetONet), a novel architecture that integrates Deep Sets principles into the DeepONet framework to address this limitation. The core innovation lies in the SetONet branch network, which processes the input function as an unordered \emph{set} of location-value pairs. This design ensures permutation invariance with respect to the input points, making SetONet inherently robust to variations in the number and locations of sensors. SetONet learns richer, spatially-aware input representations by explicitly processing spatial coordinates and function values. We demonstrate SetONet's effectiveness on several benchmark problems, including derivative/anti-derivative operators, 1D Darcy flow, and 2D elasticity. Results show that SetONet successfully learns operators under variable input sampling conditions where standard DeepONet fails. Furthermore, SetONet is architecturally robust to sensor drop-off; unlike standard DeepONet, which requires methods like interpolation to function with missing data. Notably, SetONet can achieve comparable or improved accuracy over DeepONet on fixed grids, particularly for nonlinear problems, likely due to its enhanced input representation. SetONet provides a flexible and robust extension to the neural operator toolkit, significantly broadening the applicability of operator learning to problems with variable or incomplete input data.