OPAL: Encoding Causal Understanding of Physical Systems for Robot Learning

We present OPAL (Operant Physical Agent with Language), a novel vision-language-action architecture that introduces topological constraints to flow matching for robotic control. To do so, we further introduce topological attention. Our approach models action sequences as topologically-structured representations with non-trivial constraints. Experimental results across 10 complex manipulation tasks demonstrate OPAL's superior performance compared to previous approaches, including Octo, OpenVLA, and π0. Our architecture achieves significant improvements in zero-shot performance without requiring task-specific fine-tuning, while reducing inference computational requirements by 42%. The theoretical guarantees provided by our topological approach result in more coherent long-horizon action sequences. Our results highlight the potential of constraining the search space of learning problems in robotics by deriving from fundamental physical laws, and the possibility of using topological attention to embed causal understanding into transformer architectures.