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.

We first investigate reproducing pairs in Hilbert spaces, with a focus on the discrete case. Reproducing pairs generalize frames and consist of two sequences $\Psi$ and $\Phi$, along with a bounded invertible operator $S_{\Psi,\Phi}$. The work examines sequences that are overcomplete by one element—that is, they become exact upon removal of a single element. A central result shows that if such a sequence admits a reproducing partner, the resulting exact subsequence must form a Schauder basis. This implies that systems like the Gaussian Gabor system at critical density, which lacks a Schauder basis, cannot have a reproducing partner. The result is further generalized to sequences overcomplete by finitely many elements.

Next, we introduce exponential reproducing pairs, where the sequences are weighted exponentials. The associated operator $S_{g\gamma}$ acts as a multiplication operator, and necessary and sufficient conditions are established for when a pair $(g, \gamma)$ forms an exponential reproducing pair.

Lastly, by extending a 2012 result of Heil and Yoon, we develop a two-dimensional theory for weighted exponential systems. It characterizes when weighted double exponential systems are minimal and complete, and provides necessary and sufficient conditions for exactness of arbitrary weighted systems.

Zoom Link: https://gatech.zoom.us/j/93221716846