In this paper, we revisit the Empirical Risk Minimization problem in the non-interactive local model of differential privacy.

Parameterized quantum circuits (PQCs) have emerged as a promising approach for quantum neural networks. However, understanding their expressive power in accomplishing machine learning tasks remains a crucial question. This paper investigates the expressivity of PQCs for approximating general multivariate function classes. Unlike previous Universal Approximation Theorems for PQCs, which are either nonconstructive or rely on parameterized classical data processing, we explicitly construct data re-uploading PQCs for approximating multivariate polynomials and smooth functions. We establish the first non-asymptotic approximation error bounds for these functions in terms of the number of qubits, quantum circuit depth, and number of trainable parameters. Notably, we demonstrate that for approximating functions that satisfy specific smoothness criteria, the quantum circuit size and number of trainable parameters of our proposed PQCs can be smaller than those of deep ReLU neural networks.

Predicting the distribution of future states in a stochastic system, known as belief propagation, is fundamental to reasoning under uncertainty. However, nonlinear dynamics often make analytical belief propagation intractable, requiring approximate methods. When the system model is unknown and must be learned from data, a key question arises: can we learn a model that (i) universally approximates general nonlinear stochastic dynamics, and (ii) supports analytical belief propagation? This paper establishes the theoretical foundations for a class of models that satisfy both properties. The proposed approach combines the expressiveness of normalizing flows for density estimation with the analytical tractability of Bernstein polynomials. Empirical results show the efficacy of our learned model over state-of-the-art data-driven methods for belief propagation, especially for highly non-linear systems with non-additive, non-Gaussian noise.

Density regression models allow a comprehensive understanding of data by modeling the complete conditional probability distribution. While flexible estimation approaches such as normalizing flows (NF) work particularly well in multiple dimensions, interpreting the input-output relationship of such models is often difficult, due to the black-box character of deep learning models. In contrast, existing statistical methods for multivariate outcomes such as multivariate conditional transformation models (MCTM) are restricted in flexibility and are often not expressive enough to represent complex multivariate probability distributions. In this paper, we combine MCTM with state-of-the-art and autoregressive NF to leverage the transparency of MCTM for modeling interpretable feature effects on the marginal distributions in the first step and the flexibility of neural-network-based NF techniques to account for complex and non-linear relationships in the joint data distribution. We demonstrate our method's versatility in various numerical experiments and compare it with MCTM and other NF models on both simulated and real-world data.

Understanding dissipation in open quantum systems is crucial for the development of robust quantum technologies. In this work, we introduce a Transformer-based machine learning framework to infer time-dependent dissipation rates in quantum systems governed by the Lindblad master equation. Our approach uses time series of observable quantities, such as expectation values of single Pauli operators, as input to learn dissipation profiles without requiring knowledge of the initial quantum state or even the system Hamiltonian. We demonstrate the effectiveness of our approach on a hierarchy of open quantum models of increasing complexity, including single-qubit systems with time-independent or time-dependent jump rates, two-qubit interacting systems (e.g., Heisenberg and transverse Ising models), and the Jaynes--Cummings model involving light--matter interaction and cavity loss with time-dependent decay rates. Our method accurately reconstructs both fixed and time-dependent decay rates from observable time series. To support this, we prove that under reasonable assumptions, the jump rates in all these models are uniquely determined by a finite set of observables, such as qubit and photon measurements. In practice, we combine Transformer-based architectures with lightweight feature extraction techniques to efficiently learn these dynamics. Our results suggest that modern machine learning tools can serve as scalable and data-driven alternatives for identifying unknown environments in open quantum systems.

-- Efficient real-time trajectory planning and control for fixed-wing unmanned aerial vehicles is challenging due to their non-holonomic nature, complex dynamics, and the additional uncertainties introduced by unknown aerodynamic effects. In this paper, we present a fast and efficient real-time trajectory planning and control approach for fixed-wing unmanned aerial vehicles, leveraging the differential flatness property of fixed-wing aircraft in coordinated flight conditions to generate dynamically feasible trajectories. The approach provides the ability to continuously replan trajectories, which we show is useful to dynamically account for the curvature constraint as the aircraft advances along its path. In recent years, the deployment of small Fixed-Wing Unmanned Aerial V ehicles (FW-UA Vs) has significantly increased across various applications, including environmental monitoring [1], low-altitude surveillance [2], and support for first responders in search and rescue operations [3]. Their popularity is primarily due to their superior endurance, extended operational range, and lower energy consumption compared to traditional V ertical Take-Off and Landing (VTOL) platforms like quadrotors. Since FW-UA Vs cannot hover in place or execute sharp turns and must maintain continuous motion to remain airborne, accurate trajectory planning and precise tracking are essential for their safe operations.

The optimization of geometries for aerodynamic design often relies on a large number of expensive simulations to evaluate and iteratively improve the geometries. It is possible to reduce the number of simulations by providing a starting geometry that has properties close to the desired requirements, often in terms of lift and drag, aerodynamic moments and surface areas. We show that generative models have the potential to provide such starting geometries by generalizing geometries over a large dataset of simulations. In particular, we leverage diffusion probabilistic models trained on XFOIL simulations to synthesize two-dimensional airfoil geometries conditioned on given aerodynamic features and constraints. The airfoils are parameterized with Bernstein polynomials, ensuring smoothness of the generated designs. We show that the models are able to generate diverse candidate designs for identical requirements and constraints, effectively exploring the design space to provide multiple starting points to optimization procedures. However, the quality of the candidate designs depends on the distribution of the simulated designs in the dataset. Importantly, the geometries in this dataset must satisfy other requirements and constraints that are not used in conditioning of the diffusion model, to ensure that the generated geometries are physical.