Rapid prototyping of algorithms is a critical step in modern machine learning. Most algorithms exploit linear algebra, creating a need for lightweight numerical routines which -- while potentially sub-optimal for the task at hand -- can be rapidly implemented. For the numerical solution of ill-conditioned linear systems of equations, the standard solution for prototyping is Tikhonov-regularised inversion using a nugget. However, selection of the size of nugget is often difficult, and the use of data-adaptive procedures precludes automatic differentiation, introducing instabilities into end-to-end training. Further, while data-adaptive procedures perform multiple linear solves to select the size of nugget, only the result of one such solve is returned, which we argue is wasteful. This paper aims to circumvent the above difficulties, presenting autonugget; a Python package for automatic and stable numerical solution of linear systems suitable for rapid prototyping, and fully compatible with automatic differentiation using JAX. autonugget combines multiple linear solves using Richardson extrapolation to determine the solution of the ill-conditioned system, improving in accuracy over approximations based on a single nugget.

Mask-based autoregressive models have demonstrated promising image generation capability in continuous space. However, their potential for video generation remains under-explored. In this paper, we propose VideoMAR, a concise and efficient decoder-only autoregressive image-to-video model with continuous tokens, composing temporal frame-by-frame and spatial masked generation. We first identify temporal causality and spatial bi-directionality as the first principle of video AR models, and propose the next-frame diffusion loss for the integration of mask and video generation. Besides, the huge cost and difficulty of long sequence autoregressive modeling is a basic but crucial issue. To this end, we propose the temporal short-to-long curriculum learning and spatial progressive resolution training, and employ progressive temperature strategy at inference time to mitigate the accumulation error.

Diffusion transformers (DiTs) struggle to generate images at resolutions higher than their training resolutions. The primary obstacle is that the explicit positional encodings (PE), such as RoPE, need extrapolating to unseen positions which degrades performance when the inference resolution differs from training. In this paper, We propose a Length-Extrapolatable Diffusion Transformer (LEDiT) to overcome this limitation. LEDiT needs no explicit PEs, thereby avoiding PE extrapolation. The key innovation of LEDiT lies in the use of causal attention. We demonstrate that causal attention can implicitly encode global positional information and show that such information facilitates extrapolation. We further introduce a locality enhancement module, which captures fine-grained local information to complement the global coarse-grained position information encoded by causal attention. Experimental results on both conditional and text-to-image generation tasks demonstrate that LEDiT supports up to 4 resolution scaling (e.g., from 256 256 to 512 512), achieving better image quality compared to the state-of-the-art length extrapolation methods. We believe that LEDiT marks a departure from the standard RoPE-based methods and offers a promising insight into length extrapolation.

Coarse-grained (CG) molecular simulations offer an efficient alternative to atomistic molecular dynamics to study large and complex biological systems. The accuracy of CG simulations has been increased dramatically by the introduction of machine-learned coarse-grained (MLCG) models. However, these models are typically designed to be used at a single thermodynamic point, lack temperature transferability, and can not be used to predict temperature dependent quantities like the heat capacity. Here we introduce a thermodynamically informed, temperature-transferable MLCG framework for proteins that explicitly decomposes the CG potential of mean force (PMF) into its energetic and entropic components. The model architecture enforces an exact thermodynamic relation between the energetic and entropic components of the PMF and guarantees physically consistent extrapolation and interpolation across temperature regimes. We validate this framework on an extensive dataset spanning a total of 250 $μ$s of molecular dynamics simulations across five temperatures between 300 K and 400 K for the Chignolin protein, and demonstrate that it reproduces the temperature dependency of the reference atomistic free energy surfaces, correcting the temperature-unaware baselines. Furthermore, we show that it is possible to apply an inexpensive, post-hoc temperature-dependent correction that does not require retraining the MLCG potential, accurately recovering the atomistic heat capacity at different temperatures. Overall, this work provides a physically grounded pathway toward thermodynamically transferable MLCG simulations of complex biomolecular systems.

Large language model (LLM) evaluation is increasingly costly, prompting interest in methods that speed up evaluation by shrinking benchmark datasets. Benchmark prediction (also called efficient LLM evaluation) aims to select a small subset of evaluation points and predict overall benchmark performance from that subset. In this paper, we systematically assess the strengths and limitations of 11 benchmark prediction methods across 19 diverse benchmarks. First, we identify a highly competitive baseline: Take a random sample and fit a regression model on the sample to predict missing entries. Outperforming most existing methods, this baseline challenges the assumption that careful subset selection is necessary for benchmark prediction.

Careful curation of data sources can significantly improve the performance of LLM pre-training, but predominant approaches rely heavily on intuition or costly trial-and-error, making them difficult to generalize across different data domains and downstream tasks. Although scaling laws can provide a principled and general approach for data curation, standard deterministic extrapolation from small-scale experiments to larger scales requires strong assumptions on the reliability of such extrapolation, whose brittleness has been highlighted in prior works. In this paper, we introduce a probabilistic extrapolation framework for data mixture optimization that avoids rigid assumptions and explicitly models the uncertainty in performance across decision variables. We formulate data curation as a sequential decision-making problem-multi-fidelity, multi-scale Bayesian optimization-where {data mixtures, model scale, training steps} are adaptively selected to balance training cost and potential information gain. Our framework naturally gives rise to algorithm prototypes that leverage noisy information from inexpensive experiments to systematically inform costly training decisions. To accelerate methodological progress, we build a simulator based on 472 language model pre-training runs with varying data compositions from the SlimPajama dataset. We observe that even simple kernels and acquisition functions can enable principled decisions across training models from 20M to 1B parameters and achieve 2.6x and 3.3x speedups compared to multi-fidelity BO and random search baselines. Taken together, our framework underscores potential efficiency gains achievable by developing principled and transferable data mixture optimization methods.

A transformer is merely a stack of learned data-to-data maps--yet those maps can hide rich algorithms. We train a linear, attention-only transformer on millions of masked-block completion tasks: each prompt is a masked low-rank matrix whose missing block may be (i) a scalar prediction target or (ii) an unseen kernel slice for Nyström extrapolation. The model sees only input-output pairs and a mean-squared loss; it is given no normal equations, no handcrafted iterations, and no hint that the tasks are related. Surprisingly, after training, algebraic unrolling reveals the same parameter-free update rule across all three resource regimes (full visibility, bandwidth-limited heads, rank-limited attention). We prove that this rule achieves second-order convergence on full-batch problems, cuts distributed iteration complexity, and remains accurate with compute-limited attention. Thus, a transformer trained solely to patch missing blocks implicitly discovers a unified, resource-adaptive iterative solver spanning prediction, estimation, and Nyström extrapolation--highlighting a powerful capability of in-context learning.

Diffusion transformers (DiTs) struggle to generate images at resolutions higher than their training resolutions. The primary obstacle is that the explicit positional encodings (PE), such as RoPE, need extrapolating to unseen positions which degrades performance when the inference resolution differs from training. In this paper, We propose a Length-Extrapolatable Diffusion Transformer (LEDiT) to overcome this limitation. LEDiT needs no explicit PEs, thereby avoiding PE extrapolation. The key innovation of LEDiT lies in the use of causal attention. We demonstrate that causal attention can implicitly encode global positional information and show that such information facilitates extrapolation. We further introduce a locality enhancement module, which captures fine-grained local information to complement the global coarse-grained position information encoded by causal attention. Experimental results on both conditional and text-to-image generation tasks demonstrate that LEDiT supports up to 4 resolution scaling (e.g., from 256$\times$256 to 512$\times$512), achieving better image quality compared to the state-of-the-art length extrapolation methods. We believe that LEDiT marks a departure from the standard RoPE-based methods and offers a promising insight into length extrapolation.

Modern learning systems excel at interpolation but struggle to generalize to unseen tasks outside the training distribution's support. This failure occurs even in simple settings, such as handling task parameters beyond the training range, and persists despite advances in foundation models. To this end, we develop the Relational Task Extrapolator (RTE), an algorithm designed to enable systematic extrapolation to novel tasks. The key observation is that extrapolation is inherently relational: extrapolating to unseen tasks requires learning how tasks transform into one another. If a model learns the transformation between tasks A and B during training, it can apply that same transformation to relate known tasks to unseen ones at test time. RTE operationalizes this idea by decomposing each target task into a known anchor task and a transformation linking the anchor and target. It then learns a relational operator, mapping an anchor-transformation pair to predictions for the target task. We instantiate RTE across multiple task extrapolation regimes in function prediction, e.g. where target tasks use out-of-range parameters (parameter extrapolation), have greater compositional depth (length extrapolation), and/or recombine function primitives in unseen ways (compositional extrapolation). We further extend RTE to sequence prediction, integrating it into fine-tuning algorithms for foundation models. Across empirical studies, we find that RTE substantially outperforms existing approaches on extrapolation to novel, unseen tasks.

We introduce FLUXtrapolation, a benchmark for extrapolating ecosystem fluxes under progressively harder distribution shifts. Ecosystem fluxes are central to understanding the carbon, water, and energy cycles, yet they can only be measured directly at sparsely located measurement towers. Producing global flux estimates therefore requires training models on observed sites using globally available covariates and predicting in unobserved regions, that is, upscaling. Flux upscaling is a challenging domain generalization problem that is affected by a shift in covariate distribution across climates, ecosystem types, and environmental conditions, as well as by conditional shift: important drivers remain unobserved at global scale. We provide a quantitative analysis of both these shifts in $P_X$ and $P_{Y\mid X}$. FLUXtrapolation is designed based on domain expertise on flux upscaling: it defines temporal, spatial, and temperature-based extrapolation scenarios and evaluates performance across held-out domains, temporal aggregations, and tail errors. In a pilot study, we find that baselines perform similarly under median hourly RMSE, but separate under the proposed tail-focused and multi-scale evaluation. FLUXtrapolation therefore poses a realistic and thus relevant challenge for machine learning methods under distribution shift; at the same time, progress on this benchmark would directly support the scientific goal of improving flux upscaling.