This space-bending geometry makes the equivariance to translations of convolutional networks an undesired inductive bias.

This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors defined based on advanced commutative algebra and algebraic analysis. The implementation of these priors is algorithmic and achieved using the Macaulay2 computer algebra software. An example application includes identifying the wave speed from noisy data for classical wave equations, which are widely used in physics. The method achieves high accuracy while enhancing computational efficiency.

Unlike most human-engineered systems, many biological systems rely on emergent behaviors from low-level interactions, enabling greater diversity and superior adaptation to complex, dynamic environments. This study explores emergent decentralized rotation in the Loopy multicellular robot, composed of homogeneous, physically linked, 1-degree-of-freedom cells. Inspired by biological systems like sunflowers, Loopy uses simple local interactions-diffusion, reaction, and active transport of simulated chemicals, called morphogens-without centralized control or knowledge of its global morphology. Through these interactions, the robot self-organizes to achieve coordinated rotational motion and forms lobes-local protrusions created by clusters of motor cells. This study investigates how these interactions drive Loopy's rotation, the impact of its morphology, and its resilience to actuator failures. Our findings reveal two distinct behaviors: 1) inner valleys between lobes rotate faster than the outer peaks, contrasting with rigid body dynamics, and 2) cells rotate in the opposite direction of the overall morphology. The experiments show that while Loopy's morphology does not affect its angular velocity relative to its cells, larger lobes increase cellular rotation and decrease morphology rotation relative to the environment. Even with up to one-third of its actuators disabled and significant morphological changes, Loopy maintains its rotational abilities, highlighting the potential of decentralized, bio-inspired strategies for resilient and adaptable robotic systems.

Phase association groups seismic wave arrivals according to their originating earthquakes. It is a fundamental task in a seismic data processing pipeline, but challenging to perform for smaller, high-rate seismic events which carry fundamental information about earthquake dynamics, especially with a commonly assumed inaccurate wave speed model. As a consequence, most association methods focus on larger events that occur at a lower rate and are thus easier to associate, even though microseismicity provides a valuable description of the elastic medium properties in the subsurface. In this paper, we show that association is possible at rates much higher than previously reported even when the wave speed is unknown. We propose Harpa, a high-rate seismic phase association method which leverages deep neural fields to build generative models of wave speeds and associated travel times, and first solves a joint spatio--temporal source localization and wave speed recovery problem, followed by association. We obviate the need for associated phases by interpreting arrival time data as probability measures and using an optimal transport loss to enforce data fidelity. The joint recovery problem is known to admit a unique solution under certain conditions but due to the non-convexity of the corresponding loss a simple gradient scheme converges to poor local minima. We show that this is effectively mitigated by stochastic gradient Langevin dynamics (SGLD). Numerical experiments show that \harpa~efficiently associates high-rate seismicity clouds over complex, unknown wave speeds and graciously handles noisy and missing picks.

Despite their remarkable success in approximating a wide range of operators defined by PDEs, existing neural operators (NOs) do not necessarily perform well for all physics problems. We focus here on high-frequency waves to highlight possible shortcomings. To resolve these, we propose a subfamily of NOs enabling an enhanced empirical approximation of the nonlinear operator mapping wave speed to solution, or boundary values for the Helmholtz equation on a bounded domain. The latter operator is commonly referred to as the ''forward'' operator in the study of inverse problems. Our methodology draws inspiration from transformers and techniques such as stochastic depth. Our experiments reveal certain surprises in the generalization and the relevance of introducing stochastic depth. Our NOs show superior performance as compared with standard NOs, not only for testing within the training distribution but also for out-of-distribution scenarios. To delve into this observation, we offer an in-depth analysis of the Rademacher complexity associated with our modified models and prove an upper bound tied to their stochastic depth that existing NOs do not satisfy. Furthermore, we obtain a novel out-of-distribution risk bound tailored to Gaussian measures on Banach spaces, again relating stochastic depth with the bound. We conclude by proposing a hypernetwork version of the subfamily of NOs as a surrogate model for the mentioned forward operator.