Georgia Institute of Technology College of Sciences

We develop a unified framework for improving numerical solvers with Neural Networks with Locally Converging Inputs (NNLCI). First, we applied NNLCI to 2D Maxwell’s equations with perfectly matched‐layer boundary conditions for light–PEC (perfect electric conductor) interactions. A network trained on local patches around specific PEC shapes successfully predicted solutions on globally different geometries. Next, we tested NNLCI on various ODEs: it failed for chaotic systems (e.g., double pendulum) but was effective for nonchaotic dynamics, and in simple cases can be interpreted as a well‐defined function of its inputs. Although originally formulated for hyperbolic conservation laws, NNLCI also performed well on parabolic and elliptic problems, as demonstrated in a 1D Poisson–Nernst–Planck ion‐channel model. Building on these results, we applied NNLCI to multi‐asset cash‐or‐nothing options under Black–Scholes. By correcting coarse‐ and fine‐mesh ADI solutions, NNLCI reduced RMSE by factors of 4–12 on test parameters, even when trained on a small fraction of the parameter grid. Careful treatment of far‐field boundary truncation was critical to maintain convergence far from the strike price. Finally, we demonstrate NNLCI’s first application to quantum algorithms by improving variational quantum‐algorithm (VQA) outputs for the 1D Poisson equation under realistic NISQ‐device noise. Although noisy VQA solutions deviate from classical finite‐difference references and do not converge to true solutions, NNLCI effectively maps these noisy outputs toward high‐accuracy references. We hypothesize that NNLCI implicitly composes the map from coarse quantum outputs to a noisy convergence space, then to the true solution. We discuss conditions for NNLCI to approximate a well‐defined inverse of the numerical scheme and contrast this with Monte Carlo methods, which lack deterministic intermediate states. These results establish NNLCI as a versatile, data‐efficient tool for accelerating solvers in classical and quantum settings.

The orthogonal group synchronization problem, which focuses on recovering orthogonal group elements from their corrupted pairwise measurements, encompasses examples such as high-dimensional Kuramoto model on general signed networks, $\mathbb{Z}_2$-synchronization, community detection under stochastic block models, and orthogonal Procrustes problem. The semidefinite relaxation (SDR) has proven its power in solving this problem; however, its expensive computational costs impede its widespread practical applications. We consider the Burer-Monteiro factorization approach to the orthogonal group synchronization, an effective and scalable low-rank factorization to solve large scale SDPs. Despite the significant empirical successes of this factorization approach, it is still a challenging task to understand when the nonconvex optimization landscape is benign, i.e., the optimization landscape possesses only one local minimizer, which is also global. In this work, we demonstrate that if the degree of freedom within the factorization exceeds the condition number of the ``Laplacian" (certificate matrix) at the global minimizer, the optimization landscape is absent of spurious local minima. Our main theorem is purely algebraic and versatile, and it seamlessly applies to all the aforementioned examples: the nonconvex landscape remains benign under almost identical condition that enables the success of the SDR. Finally, we will discuss the statistical sides of group synchronization by quantifying the uncertainty of both MLE and spectral estimators.

Deep learning has been widely applied and brought breakthroughs in speech recognition, computer vision, natural language processing, and many other domains. The involved deep neural network architectures and computational issues have been well studied in machine learning. But there is much less theoretical understanding about the modelling, approximation or generalization abilities of deep learning models with network architectures. An important family of structured deep neural networks is deep convolutional neural networks (CNNs) induced by convolutions. The convolutional architecture gives essential differences between deep CNNs and fully-connected neural networks, and the classical approximation theory for fully-connected networks developed around 30 years ago does not apply.  This talk describes approximation and generalization analysis of deep CNNs and related structured deep neural networks. 
 

The incipient infinite cluster was first proposed by physicists in the 1970s as a canonical example of a two-dimensional medium on which random walk is subdiffusive. It is the measure obtained in critical percolation by conditioning on the existence of an infinite cluster, which is a probability zero event. Kesten presented the first rigorous two-dimensional construction of this object as a weak limit of the one-arm event. In high dimensions, van der Hofstad and Jarai constructed the IIC as a weak limit of the two-point connection using the lace expansion. Our work presents a new high-dimensional construction which is "robust", establishing that the weak limit is independent of the choice of conditioning. The main tools used are Kesten's original two-dimensional construction combined with Kozma and Nachmias' regularity method. Our robustness allows for several applications, such as the explicit computation of the limiting distribution of the chemical distance, which forms the content of our upcoming project. This is joint work with Shirshendu Chatterjee, Jack Hanson, and Philippe Sosoe. The preprint can be found at https://arxiv.org/abs/2502.10882.