Eigen-Spike Emergence and Quadratic Equivalents for Conjugate Kernels on Nonlinearly Separable Data

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). On the one hand, these deterministic equivalents make theoretical predictions tractable by reducing a complex model to a simpler model with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful when dealing with high-dimensional nonlinearly separable data, such as performing clssification on nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a feedforward NN, under a canonical nonlinearly separable dataset, the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent to the spiked CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. In each of these scenarios, we derive a precise BBP-type phase transition in which linear classification via the CK eigenvectors becomes possible. Our analysis helps translate the power of deterministic equivalence tools in RMT to study problems of practical relevance in ML.

Unveiling Multi-regime Patterns in SciML: Distinct Failure Modes and Regime-specific Optimization

Neural networks trained under different hyperparameter settings can fall into distinct training "regimes," with consistent behavior within regimes and qualitative differences across regimes. In this paper, we study such multi-regime behavior in scientific machine learning (SciML) models through a regime-aware diagnostic framework that jointly analyzes performance, training dynamics, and loss-landscape geometry. We identify three key findings: (i) a consistent three-regime structure emerges across many standard SciML models, different constraint enforcements, and various optimizer designs; (ii) optimization effectiveness is regime-specific, with no single method performing well across all regimes; and (iii) SciML models can exhibit fine-grained failure modes that can challenge conventional interpretations of standard loss-landscape metrics. Our results provide an approach to establish a unified, task-oblivious perspective on failure modes in SciML and to inform regime-aware guidance for improving robustness. We validate these findings across widely-used SciML models, including physics-informed neural networks, neural operators, and neural ordinary differential equations, on benchmarks spanning representative ordinary and partial differential equations.

Speculative Interaction Agents: Building Real-Time Agents with Asynchronous I/O and Speculative Tool Calling

There is a growing demand for agentic AI technologies for a range of downstream applications like customer service and personal assistants. For applications where the agent needs to interact with a person, real-time low-latency responsiveness is required; for example, with voice-controlled applications, under 1 second of latency is typically required for the interaction to feel seamless. However, if we want the LLM to reason and execute an agentic workflow with tool calling, this can add several seconds or more of latency, which is prohibitive for real-time latency-sensitive applications. In our work, we propose Speculative Interaction Agents to enable real-time interaction even for agents with complex multi-turn tool calling. We propose Asynchronous I/O, which decouples the core agent reason-and-act thread from waiting for additional information from either the user or environment, thereby allowing for overlapping agentic processing while waiting on external delays. We also propose Speculative Tool Calling as a method to manage task execution when the agent is still unsure if it has received the full information or if additional user information may later be provided. For strong cloud models, our method can be applied out-of-the-box to existing real-time cloud APIs, providing 1.3-1.7$\times$ speedups with minor accuracy loss. To enable real-time interaction with small edge-scale models, we also present a clock-based training methodology that adapts the model to handle streaming inputs and asynchronous responses, and demonstrate a synthetic data generation strategy for SFT. Altogether, this approach provides 1.6-2.2$\times$ speedups with the Qwen2.5-3B-Instruct and Llama-3.2-3B-Instruct models across multiple tool calling benchmarks.

Free Decompression with Algebraic Spectral Curves

Tools from random matrix theory have become central to deep learning theory, using spectral information to provide mechanisms for modeling generalization, robustness, scaling, and failure modes. While often capable of modeling empirical behavior, practical computations are limited by matrix size, often imposing a restriction to models that are too small to be realistic. This motivates the inference of properties of larger models from the behavior of smaller ones. Free decompression (FD) is a recently proposed method for extrapolating spectral information across matrix sizes, but its utility is currently limited by strong assumptions that preclude its implementation on more realistic machine learning (ML) models. We use algebraic spectral curve theory to provide a general FD methodology for spectral densities whose Stieltjes transform satisfies an algebraic relation, a modeling assumption that is more likely to hold in practice. This recasts FD as an evolution along spectral curves which can be readily integrated. Our framework enables the expansion of spectral densities that have multiple or multi-modal bulks, that exist at multiple scales, and that contain atoms, all characteristic of real-world data and popular ML models. We demonstrate the efficacy of our framework on models of interest in modern ML, including Hessian and activation matrices associated with neural networks and large-scale diffusion models.

On Neural Scaling Laws for Weather Emulation through Continual Training

Neural scaling laws, which in some domains can predict the performance of large neural networks as a function of model, data, and compute scale, are the cornerstone of building foundation models in Natural Language Processing and Computer Vision. We study neural scaling in Scientific Machine Learning, focusing on models for weather forecasting. To analyze scaling behavior in as simple a setting as possible, we adopt a minimal, scalable, general-purpose Swin Transformer architecture, and we use continual training with constant learning rates and periodic cooldowns as an efficient training strategy. We show that models trained in this minimalist way follow predictable scaling trends and even outperform standard cosine learning rate schedules. Cooldown phases can be re-purposed to improve downstream performance, e.g., enabling accurate multi-step rollouts over longer forecast horizons as well as sharper predictions through spectral loss adjustments. We also systematically explore a wide range of model and dataset sizes under various compute budgets to construct IsoFLOP curves, and we identify compute-optimal training regimes. Extrapolating these trends to larger scales highlights potential performance limits, demonstrating that neural scaling can serve as an important diagnostic for efficient resource allocation. We open-source our code for reproducibility.

Time-Aware Prior Fitted Networks for Zero-Shot Forecasting with Exogenous Variables

In many time series forecasting settings, the target time series is accompanied by exogenous covariates, such as promotions and prices in retail demand; temperature in energy load; calendar and holiday indicators for traffic or sales; and grid load or fuel costs in electricity pricing. Ignoring these exogenous signals can substantially degrade forecasting accuracy, particularly when they drive spikes, discontinuities, or regime and phase changes in the target series. Most current time series foundation models (e.g., Chronos, Sundial, TimesFM, TimeMoE, TimeLLM, and LagLlama) ignore exogenous covariates and make forecasts solely from the numerical time series history, thereby limiting their performance. In this paper, we develop ApolloPFN, a prior-data fitted network (PFN) that is time-aware (unlike prior PFNs) and that natively incorporates exogenous covariates (unlike prior univariate forecasters). Our design introduces two major advances: (i) a synthetic data generation procedure tailored to resolve the failure modes that arise when tabular (non-temporal) PFNs are applied to time series; and (ii) time-aware architectural modifications that embed inductive biases needed to exploit the time series context. We demonstrate that ApolloPFN achieves state-of-the-art results across benchmarks, such as M5 and electric price forecasting, that contain exogenous information.

Zatom-1: A Multimodal Flow Foundation Model for 3D Molecules and Materials

General-purpose 3D chemical modeling encompasses molecules and materials, requiring both generative and predictive capabilities. However, most existing AI approaches are optimized for a single domain (molecules or materials) and a single task (generation or prediction), which limits representation sharing and transfer. We introduce Zatom-1, the first foundation model that unifies generative and predictive learning of 3D molecules and materials. Zatom-1 is a Transformer trained with a multimodal flow matching objective that jointly models discrete atom types and continuous 3D geometries. This approach supports scalable pretraining with predictable gains as model capacity increases, while enabling fast and stable sampling. We use joint generative pretraining as a universal initialization for downstream multi-task prediction of properties, energies, and forces. Empirically, Zatom-1 matches or outperforms specialized baselines on both generative and predictive benchmarks, while reducing the generative inference time by more than an order of magnitude. Our experiments demonstrate positive predictive transfer between chemical domains from joint generative pretraining: modeling materials during pretraining improves molecular property prediction accuracy.

Agentic Test-Time Scaling for WebAgents

Test-time scaling has become a standard way to improve performance and boost reliability of neural network models. However, its behavior on agentic, multi-step tasks remains less well-understood: small per-step errors can compound over long horizons; and we find that naive policies that uniformly increase sampling show diminishing returns. In this work, we present CATTS, a simple technique for dynamically allocating compute for multi-step agents. We first conduct an empirical study of inference-time scaling for web agents. We find that uniformly increasing per-step compute quickly saturates in long-horizon environments. We then investigate stronger aggregation strategies, including an LLM-based Arbiter that can outperform naive voting, but that can overrule high-consensus decisions. We show that uncertainty statistics derived from the agent's own vote distribution (entropy and top-1/top-2 margin) correlate with downstream success and provide a practical signal for dynamic compute allocation. Based on these findings, we introduce Confidence-Aware Test-Time Scaling (CATTS), which uses vote-derived uncertainty to allocate compute only when decisions are genuinely contentious. CATTS improves performance on WebArena-Lite and GoBrowse by up to 9.1% over React while using up to 2.3x fewer tokens than uniform scaling, providing both efficiency gains and an interpretable decision rule.

Learning to Discover Iterative Spectral Algorithms

We introduce AutoSpec, a neural network framework for discovering iterative spectral algorithms for large-scale numerical linear algebra and numerical optimization. Our self-supervised models adapt to input operators using coarse spectral information (e.g., eigenvalue estimates and residual norms), and they predict recurrence coefficients for computing or applying a matrix polynomial tailored to a downstream task. The effectiveness of AutoSpec relies on three ingredients: an architecture whose inference pass implements short, executable numerical linear algebra recurrences; efficient training on small synthetic problems with transfer to large-scale real-world operators; and task-defined objectives that enforce the desired approximation or preconditioning behavior across the range of spectral profiles represented in the training set. We apply AutoSpec to discovering algorithms for representative numerical linear algebra tasks: accelerating matrix-function approximation; accelerating sparse linear solvers; and spectral filtering/preconditioning for eigenvalue computations. On real-world matrices, the learned procedures deliver orders-of-magnitude improvements in accuracy and/or reductions in iteration count, relative to basic baselines. We also find clear connections to classical theory: the induced polynomials often exhibit near-equiripple, near-minimax behavior characteristic of Chebyshev polynomials.

Is Flow Matching Just Trajectory Replay for Sequential Data?

Flow matching (FM) is increasingly used for time-series generation, but it is not well understood whether it learns a general dynamical structure or simply performs an effective "trajectory replay". We study this question by deriving the velocity field targeted by the empirical FM objective on sequential data, in the limit of perfect function approximation. For the Gaussian conditional paths commonly used in practice, we show that the implied sampler is an ODE whose dynamics constitutes a nonparametric, memory-augmented continuous-time dynamical system. The optimal field admits a closed-form expression as a similarity-weighted mixture of instantaneous velocities induced by past transitions, making the dataset dependence explicit and interpretable. This perspective positions neural FM models trained by stochastic optimization as parametric surrogates of an ideal nonparametric solution. Using the structure of the optimal field, we study sampling and approximation schemes that improve the efficiency and numerical robustness of ODE-based generation. On nonlinear dynamical system benchmarks, the resulting closed-form sampler yields strong probabilistic forecasts directly from historical transitions, without training.