We develop a general framework for finding approximately-optimal preconditioners for solving linear systems.

We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterations caused by unbalanced nonlinearities, we introduce a fixed-point neural operator (FPNO) that learns the direct mapping from the current iterate to the solution by emulating fixed-point iterations. Unlike traditional line-search or trust-region algorithms, the proposed FPNO adaptively employs negative step sizes to effectively mitigate the effects of unbalanced nonlinearities. Through numerical experiments we demonstrate the computational efficiency and robustness of the proposed NP-Newton method across multiple real-world applications, especially for very strong nonlinearities.

Typical lunar landing missions involve multiple phases of braking to achieve soft-landing. The propulsion system configuration for these missions consists of throttleable engines. This configuration involves complex interconnected hydraulic, mechanical, and pneumatic components each exhibiting non-linear dynamic characteristics. Accurate modelling of the propulsion dynamics is essential for analyzing closed-loop guidance and control schemes during descent. This paper presents a learning-based system identification approach for modelling of throttleable engine dynamics using data obtained from high-fidelity propulsion model. The developed model is validated with experimental results and used for closed-loop guidance and control simulations.

Abstract--Autonomous systems increasingly rely on human feedback to align their behavior, expressed as pairwise comparisons, rankings, or demonstrations. While existing methods can adapt behaviors, they often fail to guarantee safety in safety-critical domains. We propose a safety-guaranteed, optimal, and efficient approach to solve the learning problem from preferences, rankings, or demonstrations using Weighted Signal T emporal Logic (WSTL). WSTL learning problems, when implemented naively, lead to multi-linear constraints in the weights to be learned. By introducing structural pruning and log-transform procedures, we reduce the problem size and recast the problem as a Mixed-Integer Linear Program while preserving safety guarantees. Experiments on robotic navigation and real-world Formula 1 data demonstrate that the method effectively captures nuanced preferences and models complex task objectives. Autonomous systems are increasingly part of our daily lives, from driverless cars in urban navigation to household robots performing domestic chores. Since these systems operate closely alongside humans, learning from human feedback is a natural way to ensure their behaviors align with human desires.

Safety in stochastic control systems, which are subject to random noise with a known probability distribution, aims to compute policies that satisfy predefined operational constraints with high confidence throughout the uncertain evolution of the state variables. The unpredictable evolution of state variables poses a significant challenge for meeting predefined constraints using various control methods. To address this, we present a new algorithm that computes safe policies to determine the safety level across a finite state set. This algorithm reduces the safety objective to the standard average reward Markov Decision Process (MDP) objective. This reduction enables us to use standard techniques, such as linear programs, to compute and analyze safe policies. We validate the proposed method numerically on the Double Integrator and the Inverted Pendulum systems. Results indicate that the average-reward MDPs solution is more comprehensive, converges faster, and offers higher quality compared to the minimum discounted-reward solution. Keywords: Safety Critical Systems, Robotics, Average Reward MDPs, Stochastic Control.

The Schrödinger Bridge provides a principled framework for modeling stochastic processes between distributions; however, existing methods are limited by energy-conservation assumptions, which constrains the bridge's shape preventing it from model varying-energy phenomena. To overcome this, we introduce the non-conservative generalized Schrödinger bridge (NCGSB), a novel, energy-varying reformulation based on contact Hamiltonian mechanics. By allowing energy to change over time, the NCGSB provides a broader class of real-world stochastic processes, capturing richer and more faithful intermediate dynamics. By parameterizing the Wasserstein manifold, we lift the bridge problem to a tractable geodesic computation in a finite-dimensional space. Unlike computationally expensive iterative solutions, our contact Wasserstein geodesic (CWG) is naturally implemented via a ResNet architecture and relies on a non-iterative solver with near-linear complexity. Furthermore, CWG supports guided generation by modulating a task-specific distance metric. We validate our framework on tasks including manifold navigation, molecular dynamics predictions, and image generation, demonstrating its practical benefits and versatility.

Over a decade ago, it was demonstrated that quantum computing has the potential to revolutionize numerical linear algebra by enabling algorithms with complexity superior to what is classically achievable, e.g., the seminal HHL algorithm for solving linear systems. Efficient execution of such algorithms critically depends on representing inputs (matrices and vectors) as quantum circuits that encode or implement these inputs. For that task, two common circuit representations emerged in the literature: block encodings and state preparation circuits. In this paper, we systematically study encodings matrices in the form of block encodings and state preparation circuits. We examine methods for constructing these representations from matrices given in classical form, as well as quantum two-way conversions between circuit representations. Two key results we establish (among others) are: (a) a general method for efficiently constructing a block encoding of an arbitrary matrix given in classical form (entries stored in classical random access memory); and (b) low-overhead, bidirectional conversion algorithms between block encodings and state preparation circuits, showing that these models are essentially equivalent. From a technical perspective, two central components of our constructions are: (i) a special constant-depth multiplexer that simultaneously multiplexes all higher-order Pauli matrices of a given size, and (ii) an algorithm for performing a quantum conversion between a matrix's expansion in the standard basis and its expansion in the basis of higher-order Pauli matrices.

Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these methods are scant. The rectified flow can be regarded as an approximation to optimal transport, but in contrast to other transport methods that require optimization over a function space, computing the rectified flow only requires standard statistical tools such as regression or density estimation. Because of this, one can leverage standard data analysis tools for regression and density estimation to develop empirical versions of transport maps. We study some structural properties of the rectified flow, including existence, uniqueness, and regularity, as well as the related statistical properties, such as rates of convergence and central limit theorems, for some selected estimators. To do so, we analyze separately the bounded and unbounded cases as each presents unique challenges. In both cases, we are able to establish convergence at faster rates than the ones for the usual nonparametric regression and density estimation.

Mixed-Integer Linear Programming (MILP) is a fundamental and powerful framework for modeling complex optimization problems across diverse domains. Recently, learning-based methods have shown great promise in accelerating MILP solvers by predicting high-quality solutions. However, most existing approaches are developed and evaluated in single-domain settings, limiting their ability to generalize to unseen problem distributions. This limitation poses a major obstacle to building scalable and general-purpose learning-based solvers. To address this challenge, we introduce RoME, a domain-Robust Mixture-of-Experts framework for predicting MILP solutions across domains. RoME dynamically routes problem instances to specialized experts based on learned task embeddings. The model is trained using a two-level distributionally robust optimization strategy: inter-domain to mitigate global shifts across domains, and intra-domain to enhance local robustness by introducing perturbations on task embeddings. We reveal that cross-domain training not only enhances the model's generalization capability to unseen domains but also improves performance within each individual domain by encouraging the model to capture more general intrinsic combinatorial patterns. Specifically, a single RoME model trained on three domains achieves an average improvement of 67.7% then evaluated on five diverse domains. We further test the pretrained model on MIPLIB in a zero-shot setting, demonstrating its ability to deliver measurable performance gains on challenging real-world instances where existing learning-based approaches often struggle to generalize.