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Recently introduced by some of the authors, the in-context identification paradigm aims at estimating, offline and based on synthetic data, a meta-model that describes the behavior of a whole class of systems. Once trained, this meta-model is fed with an observed input/output sequence (context) generated by a real system to predict its behavior in a zero-shot learning fashion. In this paper, we enhance the original meta-modeling framework through three key innovations: by formulating the learning task within a probabilistic framework; by managing non-contiguous context and query windows; and by adopting recurrent patching to effectively handle long context sequences. The efficacy of these modifications is demonstrated through a numerical example focusing on the Wiener-Hammerstein system class, highlighting the model's enhanced performance and scalability.

Black-box runtime verification methods for cyber-physical systems can be used to discover errors in systems whose inputs and outputs are expressed as signals over time and their correctness requirements are specified in a temporal logic. Existing methods, such as requirement falsification, often focus on finding a single input that is a counterexample to system correctness. In this paper, we study how to create test generators that can produce multiple and diverse counterexamples for a single requirement. Several counterexamples expose system failures in varying input conditions and support the root cause analysis of the faults. We present the WOGAN algorithm to create such test generators automatically. The algorithm works by training iteratively a Wasserstein generative adversarial network that models the target distribution of the uniform distribution on the set of counterexamples. WOGAN is an algorithm that trains generative models that act as test generators for runtime verification. The training is performed online without the need for a previous model or dataset. We also propose criteria to evaluate such test generators. We evaluate the trained generators on several well-known problems including the ARCH-COMP falsification benchmarks. Our experimental results indicate that generators trained by the WOGAN algorithm are as effective as state-of-the-art requirement falsification algorithms while producing tests that are as diverse as a sample from uniform random sampling. We conclude that WOGAN is a viable method to produce test generators automatically and that these test generators can generate multiple and diverse counterexamples for the runtime verification of cyber-physical systems.

We design and analyze a new paradigm for building supervised learning networks, driven only by local optimization rules without relying on a global error function. Traditional neural networks with a fixed topology are made up of identical nodes and derive their expressiveness from an appropriate adjustment of connection weights. In contrast, our network stores new knowledge in the nodes accurately and instantaneously, in the form of a lookup table. Only then is some of this information structured and incorporated into the network geometry. The training error is initially zero by construction and remains so throughout the network topology transformation phase. The latter involves a small number of local topological transformations, such as splitting or merging of nodes and adding binary connections between them. The choice of operations to be carried out is only driven by optimization of expressivity at the local scale. What we are primarily looking for in a learning network is its ability to generalize, i.e. its capacity to correctly answer questions for which it has never learned the answers. We show on numerous examples of classification tasks that the networks generated by our algorithm systematically reach such a state of perfect generalization when the number of learned examples becomes sufficiently large. We report on the dynamics of the change of state and show that it is abrupt and has the distinctive characteristics of a first order phase transition, a phenomenon already observed for traditional learning networks and known as grokking. In addition to proposing a non-potential approach for the construction of learning networks, our algorithm makes it possible to rethink the grokking transition in a new light, under which acquisition of training data and topological structuring of data are completely decoupled phenomena.

Computer programming (coding) is indispensable for researchers across disciplines, yet it remains challenging to learn and time-consuming to carry out. Generative AI, particularly large language models (LLMs), has the potential to transform coding into intuitive conversations, but best practices and effective workflows are only emerging. We dissect AI-based coding through three key lenses: the nature and role of LLMs in coding (why), six types of coding assistance they provide (what), and a five-step workflow in action with practical implementation strategies (how). Additionally, we address the limitations and future outlook of AI in coding. By offering actionable insights, this framework helps to guide researchers in effectively leveraging AI to enhance coding practices and education, accelerating scientific progress.

We develop a new computational framework to solve sequential Bayesian optimal experimental design (SBOED) problems constrained by large-scale partial differential equations with infinite-dimensional random parameters. We propose an adaptive terminal formulation of the optimality criteria for SBOED to achieve adaptive global optimality. We also establish an equivalent optimization formulation to achieve computational simplicity enabled by Laplace and low-rank approximations of the posterior. To accelerate the solution of the SBOED problem, we develop a derivative-informed latent attention neural operator (LANO), a new neural network surrogate model that leverages (1) derivative-informed dimension reduction for latent encoding, (2) an attention mechanism to capture the dynamics in the latent space, (3) an efficient training in the latent space augmented by projected Jacobian, which collectively leads to an efficient, accurate, and scalable surrogate in computing not only the parameter-to-observable (PtO) maps but also their Jacobians. We further develop the formulation for the computation of the MAP points, the eigenpairs, and the sampling from posterior by LANO in the reduced spaces and use these computations to solve the SBOED problem. We demonstrate the superior accuracy of LANO compared to two other neural architectures and the high accuracy of LANO compared to the finite element method (FEM) for the computation of MAP points and eigenvalues in solving the SBOED problem with application to the experimental design of the time to take MRI images in monitoring tumor growth. We show that the proposed computational framework achieves an amortized $180\times$ speedup.

Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient training algorithms. In this work, we integrate concepts from pullback Riemannian geometry and generative models to propose a framework for data-driven Riemannian geometry that is scalable in both geometry and learning: score-based pullback Riemannian geometry. Focusing on unimodal distributions as a first step, we propose a score-based Riemannian structure with closed-form geodesics that pass through the data probability density. With this structure, we construct a Riemannian autoencoder (RAE) with error bounds for discovering the correct data manifold dimension. This framework can naturally be used with anisotropic normalizing flows by adopting isometry regularization during training. Through numerical experiments on various datasets, we demonstrate that our framework not only produces high-quality geodesics through the data support, but also reliably estimates the intrinsic dimension of the data manifold and provides a global chart of the manifold, even in high-dimensional ambient spaces.

Semi-implicit spectral deferred correction (SDC) methods provide a systematic approach to construct time integration methods of arbitrarily high order for nonlinear evolution equations including conservation laws. They converge towards $A$- or even $L$-stable collocation methods, but are often not sufficiently robust themselves. In this paper, a family of SDC methods inspired by an implicit formulation of the Lax-Wendroff method is developed. Compared to fully implicit approaches, the methods have the advantage that they only require the solution of positive definite or semi-definite linear systems. Numerical evidence suggests that the proposed semi-implicit SDC methods with Radau points are $L$-stable up to order 11 and require very little diffusion for orders 13 and 15. The excellent stability and accuracy of these methods is confirmed by numerical experiments with 1D conservation problems, including the convection-diffusion, Burgers, Euler and Navier-Stokes equations.

We provide a tight asymptotic characterization of the error exponent for classical-quantum channel coding assisted by activated non-signaling correlations. Namely, we find that the optimal exponent\, -- \,also called reliability function\, -- \,is equal to the well-known sphere packing bound, which can be written as a single-letter formula optimized over Petz-R\'enyi divergences. Remarkably, there is no critical rate and as such our characterization remains tight for arbitrarily low rates below the capacity. On the achievability side, we further extend our results to fully quantum channels. Our proofs rely on semi-definite program duality and a dual representation of the Petz-R\'enyi divergences via Young inequalities. As a result of independent interest, we find that the Petz-R\'enyi divergences of order $\alpha\in[0,2]$ are upper bounded by the sandwiched R\'enyi divergences of order $1/(2-\alpha)\in[1/2,\infty]$.

We propose a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green--Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.