The computational efficiency of many neural operators, widely used for learning solutions of PDEs, relies on the fast Fourier transform (FFT) for performing spectral computations. As the FFT is limited to equispaced (rectangular) grids, this limits the efficiency of such neural operators when applied to problems where the input and output functions need to be processed on general non-equispaced point distributions. Leveraging the observation that a limited set of Fourier (Spectral) modes suffice to provide the required expressivity of a neural operator, we propose a simple method, based on the efficient direct evaluation of the underlying spectral transformation, to extend neural operators to arbitrary domains. An efficient implementation of such direct spectral evaluations is coupled with existing neural operator models to allow the processing of data on arbitrary non-equispaced distributions of points. With extensive empirical evaluation, we demonstrate that the proposed method allows us to extend neural operators to arbitrary point distributions with significant gains in training speed over baselines, while retaining or improving the accuracy of Fourier neural operators (FNOs) and related neural operators.

We present Aquarium, a differentiable fluid-structure interaction solver for robotics that offers stable simulation, accurately coupled fluid-robot physics in two dimensions, and full differentiability with respect to fluid and robot states and parameters. Aquarium achieves stable simulation with accurate flow physics by directly integrating over the incompressible Navier-Stokes equations using a fully implicit Crank-Nicolson scheme with a second-order finite-volume spa-tial discretization. The fluid and robot physics are coupled using the immersed-boundary method by formulating the no-slip condition as an equality constraint applied directly to the Navier-Stokes system. This choice of coupling allows the fluid-structure interaction to be posed and solved as a nonlinear optimization problem. This optimization-based formulation is then exploited using the implicit-function theorem to compute derivatives. Derivatives can then be passed to downstream gradient-based optimization or learning algorithms. We demon-strate Aquarium's ability to accurately simulate coupled fluid-robot physics with numerous 2D examples, including a cylinder in free stream and a soft robotic fish tail with hardware validation. We also demonstrate Aquarium's ability to provide analytical gradients by performing gradient-based shape-and-gait optimization of an oscillating diamond foil to maximize its generated thrust.

Aquatic locomotion is a classic fluid-structure interaction (FSI) problem of interest to biologists and engineers. Solving the fully coupled FSI equations for incompressible Navier-Stokes and finite elasticity is computationally expensive. Optimizing robotic swimmer design within such a system generally involves cumbersome, gradient-free procedures on top of the already costly simulation. To address this challenge we present a novel, fully differentiable hybrid approach to FSI that combines a 2D direct numerical simulation for the deformable solid structure of the swimmer and a physics-constrained neural network surrogate to capture hydrodynamic effects of the fluid. For the deformable solid simulation of the swimmer’s body, we use state-of-the-art techniques from the field of computer graphics to speed up the finite-element method (FEM). For the fluid simulation, we use a U-Net architecture trained with a physics-based loss function to predict the flow field at each time step. The pressure and velocity field outputs from the neural network are sampled around the boundary of our swimmer using an immersed boundary method (IBM) to compute its swimming motion accurately and efficiently. We demonstrate the computational efficiency and differentiability of our hybrid simulator on a 2D carangiform swimmer. Due to differentiability, the simulator can be used for computational design of controls for soft bodies immersed in fluids via direct gradient-based optimization.

Soft robotics has the potential to revolutionize robotic locomotion, in particular, soft robotic swimmers offer a minimally invasive and adaptive solution to explore and preserve our oceans. Unfortunately, current soft robotic swimmers are vastly inferior to evolved biological swimmers, especially in terms of controllability, efficiency, maneuverability, and longevity. Additionally, the tedious iterative fabrication and empirical testing required to design soft robots has hindered their optimization. In this work, we tackle this challenge by providing an efficient and straightforward pipeline for designing and fabricating soft robotic swimmers equipped with electrostatic actuation. We streamline the process to allow for rapid additive manufacturing, and show how a differentiable simulation can be used to match a simplified model to the real deformation of a robotic swimmer. We perform several experiments with the fabricated swimmer by varying the voltage and actuation frequency of the swimmer's antagonistic muscles. We show how the voltage and frequency vary the locomotion speed of the swimmer while moving in liquid oil and observe a clear optimum in forward swimming speed. The differentiable simulation model we propose has various downstream applications, such as control and shape optimization of the swimmer; optimization results can be directly mapped back to the real robot through our sim-to-real matching.