arXiv subjects

Bing-Ze Lu

Publications and source records attributed to Bing-Ze Lu.

6 recordsLinked to original sources

arXiv2026

A multilevel stochastic-gradient neural solver for boundary integral equations

We propose a multilevel stochastic-gradient neural solver (MLSG) for second-kind boundary integral equations. MLSG represents the unknown boundary density using a neural network optimized by stochastic residual minimization over a hierarchy of successively refined Nystr\"om discretizations. Upon transitioning from one level to the next, the network parameters obtained on the previous level initialize training on the current one. This coarse-to-fine strategy retains a continuous, grid-independent density representation and is designed to reduce the total computational effort required to reach a prescribed residual tolerance at the target resolution. The algorithm avoids grid-transfer operators and hierarchical fast-summation machinery, relying instead on batched kernel evaluations and standard network forward and backward passes that map efficiently onto modern GPU architectures. For uniformly stable second-kind discretizations, so strongly nonuniform contraction rates originate in the empirical neural tangent kernel (NTK) rather than in the discretized operator. Within each level, parameter updates can reshape the NTK, while refinement re-samples the tangent kernel on a richer discrete space and reveals directions that were not adequately resolved on coarser levels. A cross-level estimate bounds the warm-start loss in terms of the preceding training tolerance and quadrature error, motivating a tolerance schedule that balances optimization and discretization errors. Experiments on Laplace/Poisson and Helmholtz problems in two and three dimensions, together with an exterior Robin problem on a hypersurface in $\R^4$, demonstrate the method under both parametric and signed-distance surface representations at up to million-scale discretizations.

math.NA↗

arXiv2026

Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks

This paper studies the enumeration of seven subclasses of level-$2$ phylogenetic networks under various planarity and structural constraints, including terminal planar, tree-child, and galled networks. We derive their exponential generating functions, recurrence relations, and asymptotic formulas. Specifically, we show that the number of networks of size $n$ in each class follows: \[ N_n \sim c \cdot n^{n-1} \cdot \gamma^n, \] where $c$ is a class-specific constant and $\gamma$ is the corresponding growth rate. Our results reveal that being terminal planar can significantly reduce the growth rate of general level-2 networks, but has only a minor effect on the growth rates of tree-child and galled level-2 networks. Notably, the growth rate of 3.83 for level-$2$ terminal planar galled tree-child networks is remarkably close to the rate of 2.94 for level-$1$ networks.

math.CO↗

arXiv2025

Artifacts of Numerical Integration in Learning Dynamical Systems

In many applications, one needs to learn a dynamical system from its solutions sampled at a finite number of time points. The learning problem is often formulated as an optimization problem over a chosen function class. However, in the optimization procedure, prediction data from generic dynamics requires a numerical integrator to assess the mismatch with the observed data. This paper reveals potentially serious effects of a chosen numerical scheme on the learning outcome. Specifically, the analysis demonstrates that a damped oscillatory system may be incorrectly identified as having "anti-damping" and exhibiting a reversed oscillation direction, even though it adequately fits the given data points. This paper shows that the stability region of the selected integrator will distort the nature of the learned dynamics. Crucially, reducing the step size or raising the order of an explicit integrator does not, in general, remedy this artifact, because higher-order explicit methods have stability regions that extend further into the right half complex plane. Furthermore, it is shown that the implicit midpoint method can preserve either conservative or dissipative properties from discrete data, offering a principled integrator choice even when the only prior knowledge is that the system is autonomous.

math.NA↗

arXiv2025

Optimizing Mixed Quantum Channels via Projected Gradient Dynamics

Designing a mixed quantum channel is challenging due to the complexity of the transformations and the probabilistic mixtures of more straightforward channels involved. Fully characterizing a quantum channel generally requires preparing a complete set of input states, such as a basis for the state space, and measuring the corresponding output states. In this work, we begin by investigating a single input-output pair using projected gradient dynamics. This approach applies optimization flows constrained to the Stiefel manifold and the probabilistic simplex to identify the original quantum channel. The convergence of the flow is guaranteed by its relationship to the Zariski topology. We present numerical investigations of models adapted to various scenarios, including those with multiple input-output pairs, highlighting the flexibility and efficiency of our proposed method.

math.NA↗

arXiv2025

An Iterative Methodology for Unitary Quantum Channel Search

In this paper, we propose an iterative algorithm using polar decomposition to approximate a channel characterized by a single unitary matrix based on input-output quantum state pairs. In limited data, we state and prove that the optimal solution obtained from our method using one pair with a specific structure will generate an equivalent class, significantly reducing the dimension of the searching space. Furthermore, we prove that the unitary matrices describing the same channel differ by a complex number with modulus 1. We rigorously prove our proposed algorithm can ultimately identify a critical point, which is also a local minimum of the established objective function.

math.NA↗

arXiv2023

Nurse Scheduling Problem via PyQUBO

The nurse scheduling problem is a critical optimization challenge in healthcare management. It aims to balance staffing demands, nurse satisfaction, and patient care quality. Corresponding to the constraints inherent in this scheduling problem, we detail the mathematical formulation step-by-step. We then utilize a quantum-inspired technique, the simulated annealing algorithm, and a quadratic unconstrained binary optimization model to optimize workload and increase nurse preferences. Numerical experiments are implemented to show the capacity of our proposed techniques. Our findings indicate a promising direction for future research, with potential applications extending beyond nurse scheduling to other complex optimization problems.

math.OC↗