DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning

In the realm of computational physics, an enduring topic is the numerical solutions to partial differential equations (PDEs). Recently, the attention of researchers has shifted towards Neural Operator methods, renowned for their capability to approximate ``operators'' -- mappings from functions to functions. Despite the universal approximation theorem within neural operators, ensuring error bounds often requires employing numerous Fourier layers. However, what about lightweight models? In response to this question, we introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis. To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers, enhancing their ability to handle sum-of-products structures inherent in many physical systems. Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets. Furthermore, by analyzing Fourier components' weights, we can symbolically discern the physical significance of each term. This sheds light on the opaque nature of neural networks, unveiling underlying physical principles.

Safe Non-Stochastic Control of Control-Affine Systems: An Online Convex Optimization Approach

We study how to safely control nonlinear control-affine systems that are corrupted with bounded non-stochastic noise, i.e., noise that is unknown a priori and that is not necessarily governed by a stochastic model. We focus on safety constraints that take the form of time-varying convex constraints such as collision-avoidance and control-effort constraints. We provide an algorithm with bounded dynamic regret, i.e., bounded suboptimality against an optimal clairvoyant controller that knows the realization of the noise a prior. We are motivated by the future of autonomy where robots will autonomously perform complex tasks despite real-world unpredictable disturbances such as wind gusts. To develop the algorithm, we capture our problem as a sequential game between a controller and an adversary, where the controller plays first, choosing the control input, whereas the adversary plays second, choosing the noise's realization. The controller aims to minimize its cumulative tracking error despite being unable to know the noise's realization a prior. We validate our algorithm in simulated scenarios of (i) an inverted pendulum aiming to stay upright, and (ii) a quadrotor aiming to fly to a goal location through an unknown cluttered environment.