Neural operators provide a framework for learning solution operators of partial differential equations (PDEs), enabling efficient surrogate modeling for complex systems. While universal approximation results are now well understood, approximation analysis specific to nonlinear reaction-diffusion systems remains limited. In this paper, we study neural operators applied to the solution mapping from initial conditions to time-dependent solutions of a generalized Gierer-Meinhardt reaction-diffusion system, a prototypical model of nonlinear pattern formation. Our main results establish explicit approximation error bounds in terms of network depth, width, and spectral rank by exploiting the Laplacian spectral representation of the Green's function underlying the PDE. We show that the required parameter complexity grows at most polynomially with respect to the target accuracy, demonstrating that Laplacian eigenfunction-based neural operator architectures alleviate the curse of parametric complexity encountered in generic operator learning. Numerical experiments on the Gierer-Meinhardt system support the theoretical findings.

Vision-language grounding is a fundamental and crucial problem in multi-modal understanding.

Robotic manipulation systems are increasingly deployed across diverse domains. Y et existing multi-modal learning frameworks lack inherent guarantees of geometric consistency, struggling to handle spatial transformations such as rotations and translations. While recent works attempt to introduce equivariance through bespoke architectural modifications, these methods suffer from high implementation complexity, computational cost, and poor portability. Inspired by human cognitive processes in spatial reasoning, we propose Eq.Bot, a universal canonicalization framework grounded in SE(2) group eq uivariant theory for robot ic manipulation learning. Our framework transforms observations into a canonical space, applies an existing policy, and maps the resulting actions back to the original space. As a model-agnostic solution, Eq.Bot aims to endow models with spatial equivariance without requiring architectural modifications. Extensive experiments demonstrate the superiority of Eq.Bot under both CNN-based (e.g., CLI-Port) and Transformer-based (e.g., OpenVLA-OFT) architectures over existing methods on various robotic manipulation tasks, where the most significant improvement can reach 50.0%.

Batteryless systems frequently face power failures, requiring extra runtime buffers to maintain inference progress and leaving only a memory space for storing ultra-tiny deep neural networks (DNNs). Besides, making these models responsive to stochastic energy harvesting dynamics during inference requires a balance between inference accuracy, latency, and energy overhead. Recent works on compression mostly focus on time and memory, but often ignore energy dynamics or significantly reduce the accuracy of pre-trained DNNs. Existing energy-adaptive inference works modify the architecture of pre-trained models and have significant memory overhead. Thus, energy-adaptive and accurate inference of pre-trained DNNs on batteryless devices with extreme memory constraints is more challenging than traditional microcontrollers. We combat these issues by proposing FreeML, a framework to optimize pre-trained DNN models for memory-efficient and energy-adaptive inference on batteryless systems. FreeML comprises (1) a novel compression technique to reduce the model footprint and runtime memory requirements simultaneously, making them executable on extremely memory-constrained batteryless platforms; and (2) the first early exit mechanism that uses a single exit branch for all exit points to terminate inference at any time, making models energy-adaptive with minimal memory overhead. Our experiments showed that FreeML reduces the model sizes by up to $95 \times$, supports adaptive inference with a $2.03-19.65 \times$ less memory overhead, and provides significant time and energy benefits with only a negligible accuracy drop compared to the state-of-the-art.