The requirement of generating predictions that exactly fulfill the fundamental symmetry of the corresponding physical quantities has profoundly shaped the development of machine-learning models for physical simulations. In many cases, models are built using constrained mathematical forms that ensure that symmetries are enforced exactly. However, unconstrained models that do not obey rotational symmetries are often found to have competitive performance, and to be able to \emph{learn} to a high level of accuracy an approximate equivariant behavior with a simple data augmentation strategy. In this paper, we introduce rigorous metrics to measure the symmetry content of the learned representations in such models, and assess the accuracy by which the outputs fulfill the equivariant condition. We apply these metrics to two unconstrained, transformer-based models operating on decorated point clouds (a graph neural network for atomistic simulations and a PointNet-style architecture for particle physics) to investigate how symmetry information is processed across architectural layers and is learned during training. Based on these insights, we establish a rigorous framework for diagnosing spectral failure modes in ML models. Enabled by this analysis, we demonstrate that one can achieve superior stability and accuracy by strategically injecting the minimum required inductive biases, preserving the high expressivity and scalability of unconstrained architectures while guaranteeing physical fidelity.

This story appears in the April 2026 print edition. While some stories from this issue are not yet available to read online, you can explore more from the magazine . Get our editors' guide to what matters in the world, delivered to your inbox every weekday. The race to power AI is already remaking the physical world. Three Mile Island's cooling towers have until recently served as grave markers for America's nuclear-power industry. A s we drove through southwest Memphis, KeShaun Pearson told me to keep my window down--our destination was best tasted, not viewed. Along the way, we passed an abandoned coal plant to our right, then an active power plant to our left, equipped with enormous natural-gas turbines. Pearson, who directs the nonprofit Memphis Community Against Pollution, was bringing me to his hometown's latest industrial megaproject.

Minamitorishima is nearly 1,250 miles east of Tokyo. The island is surrounded by a coral atoll and is only 0.6 miles wide. Breakthroughs, discoveries, and DIY tips sent six days a week. Nuclear power is on the rise around the world, but with it comes an extremely pressing question: where will all of the radioactive waste be stored? For Japan, one answer may lie in literally the most remote location at their disposal.

Spatio-temporal forecasting is fundamental to intelligent systems in transportation, climate science, and urban planning. However, training deep learning models on the massive, often redundant, datasets from these domains presents a significant computational bottleneck. Existing solutions typically focus on optimizing model architectures or optimizers, while overlooking the inherent inefficiency of the training data itself. This conventional approach of iterating over the entire static dataset each epoch wastes considerable resources on easy-to-learn or repetitive samples. In this paper, we explore a novel training-efficiency techniques, namely learning from complexity with dynamic sample pruning, ST-Prune, for spatio-temporal forecasting. Through dynamic sample pruning, we aim to intelligently identify the most informative samples based on the model's real-time learning state, thereby accelerating convergence and improving training efficiency. Extensive experiments conducted on real-world spatio-temporal datasets show that ST-Prune significantly accelerates the training speed while maintaining or even improving the model performance, and it also has scalability and universality.

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Our experimental results demonstrate that the ranking algorithm-based classifiers provide plans that enhance infection rate control, especially with increased action budgets for small networks.

More recently, Newton-type methods using sparsified Hessian matrices have demonstrated promising results on OT computation, but there still remain a lot of unresolved open questions.