#01Nov 10, 2026
cs.LG
Discovering Equations for the Thermodynamics of South Asian Monsoon Precipitation
Savannah L. Ferretti, Jane W. Baldwin, Michael S. Pritchard and 1 more
South Asian monsoon precipitation depends on the thermodynamic structure of the lower troposphere, but how its ingredients combine to shape precipitation remains unclear, partly because this dependence is nonlocal in the vertical. We use symbolic distillation, which combines neural networks and symbolic regression, to discover interpretable precipitation equations from 21 years of June-August reanalysis data. Neural networks first learn the vertical thermodynamic structure most relevant to precipitation, condensing each variable into a single interpretable feature not tied to a particular vertical grid. Then, symbolic regression derives closed-form equations from these features and the surface state. Our best equation explains 1.5 times as much precipitation variance as a widely used buoyancy-based diagnostic and recovers about 85% of neural network skill while using about a thousand times fewer parameters. Unlike the buoyancy diagnostic, which adds moisture and instability, the equations treat them as competing controls, with the heaviest precipitation occurring when the lower free troposphere is humid. Over ocean, sensible heating is associated with additional precipitation mainly when near-surface air is warm and moist, improving predicted precipitation and its diurnal cycle. Although the learned features were optimized only for precipitation prediction, the equations largely respond to each feature as expected from convective physics, consistent with the reanalysis. These responses reduce the dependence of monsoon precipitation on the entire vertical column to a few simple, testable constraints that could guide data-driven precipitation models to make them more physically consistent.
#02Nov 10, 2026
math.AP
Bayesian inverse problems and epidemic-threshold uncertainty for a heterogeneous SIS reaction-diffusion model
Duc-Lam Duong
We study Bayesian inference of persistence from early observations of a heterogeneous susceptible--infected--susceptible (SIS) reaction--diffusion model with standard incidence. We show that the leading prevalence dynamics depend on the unknown coefficients only through the difference between the transmission and recovery rates and the infected diffusivity. The same pair determines the principal eigenvalue whose sign governs persistence, so early prevalence can inform the threshold even when the individual rates remain uncertain. We prove global weak well-posedness and quantitative parameter stability for the SIS system and use the resulting observation-map estimates to establish Bayesian well-posedness. For an initial infection of size $\varepsilon$, we derive a uniform second-order expansion that describes how the separate rates enter prevalence. Incidence depends explicitly on transmission at leading order. When the noise standard deviation is proportional to $\varepsilon$, we prove $O(\varepsilon)$ convergence in Hellinger distance of the nonlinear posteriors to their low-prevalence limits for both prevalence alone and joint prevalence and incidence data. We bound the principal eigenvalue using the growth rate of the infected $L^2$ norm between two times. This rate gives a lower bound for the full nonlinear system, and the upper bound separates spatial transients from nonlinear saturation. Uniform spectral estimates and bounds on infection size and measurement error give sufficient conditions for persistence and extinction. We also bound the error in posterior persistence probabilities under forward and eigenvalue approximation, accounting for posterior mass near the threshold. Numerical examples illustrate threshold inference and the predicted \(O(\varepsilon)\) convergence of the nonlinear posteriors to their low-prevalence limits in Hellinger distance.
#03Nov 10, 2026
stat.ML
A Theory of Cross-Validation Variance
Nathaniel S. O'Connell, Jaime L. Speiser
Cross-validation estimates predictive performance by repeatedly splitting the data into training and validation sets, fitting the model on the first and evaluating it on the second. Every observation influences the estimate in two roles, through its own evaluation when it is held out and through the models it helps fit when it trains. The random splits determine its weight in each role. From this representation, we prove an exact finite-sample identity for the variance of the cross-validated estimate. For a given splitting scheme with a common training set size, the variance is a function of these weights, with constants of the model, the metric, and the population. It covers metrics averaged over observations, such as mean squared error, and over case--control pairs, such as the AUC. The identity shows which schemes to use. The variance decomposes into sampling variance, which depends only on the training size, and variance due to the random fold splitting, which arises when the weights change from one draw of the splits to the next. K-fold cross-validation, whose weights are equal across observations by design, is therefore variance optimal at first order; Monte Carlo and bootstrap cross-validation are not. The first-order variance of K-fold does not depend on K, and under reasonable conditions on the model leave-one-out has the smallest variance of any K-fold scheme. For the AUC the same holds for leave-pair-out, which evaluates every case--control pair. Finally, we show the cross-validated estimate is asymptotically normal with the identity's full variance. Together, these results provide a general theoretical foundation for inference on cross-validated performance and establish principles for selecting variance-optimal cross-validation designs.
#04Nov 10, 2026
cs.RO
One Policy for 100 Household Tasks: Our Solution for the 2026 BEHAVIOR Challenge
Ilia Larchenko, Gleb Zarin, Atindra Nair and 4 more
We describe our solution for the 2026 BEHAVIOR Challenge: 100 long-horizon household tasks in the OmniGibson simulator. We placed 9th on the final leaderboard, the only independent team in the top 10. The core of the solution is a single vision-language-action (VLA) model, one checkpoint for all 100 tasks, built on our 1st-place 2025 solution, which is based on $π_{0.5}$. On the 50 tasks shared with 2025, its Q rises from 0.260 to 0.371; across all 100 tasks it scores 0.392. Our main conclusion from 2025 was that the model was undertrained, so most of the 2026 work went into training the model for longer, optimizing some training parameters, making each training step slightly faster and reducing the gradient noise: a migration from 50 to 100 tasks (semantic stages built from the demonstrations' skill annotations, task and stage embeddings of new tasks warm-started from similar old ones, and a 21-dimensional action space matching the teleoperation controls); variance-reduced multi-sample flow matching with a simple noise-vs-compute model that picks the number of action-expert samples per observation without ablations; and a 200k-step training run, about one pass over the 211M frames of the demonstrations. A cloud evaluation infrastructure keeps a shared record with provenance, episode diagnostics and restorable simulator states, and runs six simulators on one GPU together faster than real time. On top of the learned policy, 14 tasks are routed to a GPT-6 Astra coding agent that controls the robot through observation and motion tools, starting every episode from a frozen copy of its own practice session on training instances. Across these 14 tasks, Astra's mean Q is 0.639, against 0.108 for the learned policy. The public submission included only part of the Astra results and scored 0.446 Q with 27.3% full-task success.
#05Nov 10, 2026
cs.LG
TorchCurves: Differentiable Parametric Curves
Alex Shtoff
We present TorchCurves, a PyTorch library that treats vector-valued parametric curves as learnable primitives. Its central operation maps batches of scalar inputs to points on curves represented in B-spline or Legendre polynomial bases, with differentiation through both inputs and coefficients. Curve coefficients can be optimized jointly with the surrounding model or produced by another network. The library provides trainable modules, reusable basis modules, and a functional interface, supporting applications ranging from continuous feature embeddings to Kolmogorov-Arnold networks and functional regression. Its implementation combines classical evaluation algorithms with vectorized PyTorch operations, emphasizing readability and maintainability without custom hardware kernels. Timing experiments compare curve evaluation against NumPy and SciPy and KAN training against EfficientKAN across a range of problem sizes. The results illustrate where this generality remains computationally practical and where overhead is substantial. Library is available on GitHub at https://github.com/alexshtf/torchcurves.