We introduce FLASH-MAX, a shallow, exact-by-construction neural network architecture for predicting homogeneous electromagnetic fields from sparse pointwise observations. Each hidden neuron represents a separate exact solution to Maxwell's equations, so that the network satisfies the governing equations symbolically by construction and can be trained end-to-end from sparse data within seconds. We prove a universal approximation result showing that this exact model class remains universal on arbitrary domains. FLASH-MAX reaches sub-1% relative validation error from about 1K sparse pointwise observations in seconds, all while maintaining a zero PDE residual, and keeps single-digit errors even for only 100 observations sampled from 3D space. These results suggest that moving governing structure from the loss into the hypothesis class can dramatically improve the trade-off between precision and optimization speed in scientific machine learning.

Providing non-conservative uncertainty quantification for function estimates derived from noisy observations remains a fundamental challenge in statistical machine learning, particularly for applications in safety-critical domains. In this work, we propose novel non-asymptotic probabilistic uniform error bounds for kernel-based regression. Compared to related bounds in the literature that are restricted to (conditionally) independent sub-Gaussian noise, our bounds allow to consider a broad class of non-Gaussian distributions, such as sub-Gaussian, bounded, sub-exponential, and variance/moment-bounded noise. Moreover, our results apply to correlated and uncorrelated noise. We compare our proposed error bounds with existing results in terms of the induced uncertainty region and their performance in safe control, demonstrating the tightness of the proposed bounds.

Leveraging the compositional nature of our world to expedite learning and facilitate generalization is a hallmark of human perception. In machine learning, on the other hand, achieving compositional generalization has proven to be an elusive goal, even for models with explicit compositional priors. To get a better handle on compositional generalization, we here approach it from the bottom up: Inspired by identifiable representation learning, we investigate compositionality as a property of the data-generating process rather than the data itself. This reformulation enables us to derive mild conditions on only the support of the training distribution and the model architecture, which are sufficient for compositional generalization. We further demonstrate how our theoretical framework applies to real-world scenarios and validate our findings empirically. Our results set the stage for a principled theoretical study of compositional generalization.

It consists of an image encoder with a Vision Transformer [17] architecture, a text encoder with a similar Transformer architecture, and heads that predict bounding boxes and label scores from provided images and text queries. Input(s) An image and a list of free-text object descriptions (queries).

A.1 Preliminary Study2 The basic GPT-2 model1 is trained from scratch on each corpus, which has 12 transformer blocks3 and 12 attention heads with 768 hidden dimensions. The Huggingface transformers [4] and Pytorch4 toolkit [2] are used to train the GPT-2 model in the distributed manner on A100 GPU server. The5 hyper-parameters during training are shown in Table 1.6 Hyper-parameter Value Optimization steps 100K Test interval 10K Dropout rate 0.1 Grad clipping 1.0 Learning rate 5e 5 Batch size 128 Maximum sequence length 256 Warmup steps 10K Learning scheduler Linear decay Random seed 0 Number of GPUs 4 Learning objective Cross-Entropy Loss Table 1: The hyper-parameters during GPT-2 training procedure. Most of the hyper-parameters for our proposed method are the same as that in Table 1 for better8 variable controlling. The specific hyper-parameters for our proposed method are the length of9 repetitive n-gram and its repetition dropout rate p, which are set as 2 and 0.6, respectively.10

This paper solves three NP-hard routing problems, traveling salesman problem (TSP), prize collecting TSP (PCTSP), and capacitated vehicle routing problem (CVRP). This section provides detailed descriptions of PCTSP and CVRP (for TSP, see section 3). The PCTSP is similar to TSP, while there are differences in that we do not have to visit all the nodes and that the destination is not the first node but the depot node, i.e., a tour is not a cycle. Let N be the number of nodes. The problem instance of PCTSP is s = {(xi,λi,µi)}N+1i=1, where the xi R2 is in 2D euclidean coordinates, λi R is the penalty of unvisited node, and µi R is the prize of visited node. The L(π|s) is the tour length, and λ(π|s) is the total penalty of the unvisited nodes.