Title: Self-supervised In-context Operator Learning for Stochastic Mean-Field Control

Abstract

Stochastic mean-field control coordinates large populations of interacting agents under noise, from swarm planning and systemic risk to Schrödinger bridges. Classical solvers discretize the Fokker-Planck equation on a mesh, so their cost grows exponentially with the state dimension and a tensor-product grid is already out of reach past two dimensions. Deep neural network solvers lift that ceiling but still handle one instance per run, re-optimizing whenever the initial law, the target, or a cost weight changes. This talk presents a mesh-free solution operator for a whole family of such problems, in which a task enters as a prompt and the optimal transport map comes out in one forward pass. We pass to the probability-flow ODE and parameterize its flow map by a prompt-conditioned coupling flow whose exact inverse and analytic log-determinant give the score at linear cost in the dimension, against the cubic cost of a generic map.

Title: Training-Free Universal Approximation by Prompting Random Transformers

Abstract

How expressive is prompting a transformer? Answering this question is important for separating the roles of prompting, architecture, and pretraining in transformer models, and for determining whether task-specific behavior must be stored in model weights or can instead be induced at inference time through the prompt. We show, in an approximation-theoretic sense, that pretraining is not strictly necessary: a single-layer softmax attention network with random, untrained weights can approximate any Hölder function on a compact manifold when steered by an appropriate soft prompt. Guided by the connection between softmax attention and kernel methods, we construct explicit soft prompts—a prompt per target function, independent of the query—as solutions to linear systems matching attention logits to Gaussian kernel exponents, under which the frozen transformer emulates the classical Nadaraya-Watson kernel estimator. The construction requires only a mild rank condition on the weights, which we show holds almost surely under Gaussian initialization. The prompted network inherits the theoretical guarantees of kernel regression, leading to universal approximation theorems with minimax-optimal rates that depend on the intrinsic dimension. We further quantify the cost of prompting, exposing a tradeoff between the norm of the constructed soft prompt tokens, prompt length, and hidden dimension.

Title: Outpainting: spatially extending aero-optic phase screens

Abstract

Aero-optic effects distort light wave propagation near a high-speed aircraft, thereby degrading performance in airborne imaging and communication systems. Measuring aero-optic data through experiment is costly and the resulting data often has a limited spatial size. Further, alternative methods for simulating this data, including computational fluid dynamics and conventional phase screen generation algorithms (e.g., boiling flow), face drawbacks such as large computation time or inaccurate statistics. More recently, data-driven algorithms have been proposed that can synthesize data that matches relevant statistics of measured aero-optic data. However, these methods cannot spatially extend aero-optic data. In this paper, we introduce ReVAR-ext (Re-whitened Vector AutoRegression-extender), an algorithm that builds on an existing data-driven approach, ReVAR, to spatially extend measured aero-optic data (a process called outpainting) and match the spatial and temporal correlations of the measured data. ReVAR-ext generalizes the generation process of ReVAR by combining multiple sets of synthetic data with the input measured data. This approach generates multiple fixed-sized synthetic images, each of which overlaps with the input data, and then stitches them together. When paired with ReVAR, the ReVAR-ext algorithm can generate aero-optic data with arbitrary temporal duration and arbitrary spatial size. Our experiments show that extended data generated by ReVAR-ext closely matches the temporal power spectrum of two measured aero-optic data sets. Further, the extended data approximately matches the spatial autocorrelation, with reduced accuracy at large spatial lags and at vertical lags.

Title: TBA

Abstract

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