arXiv subjects

Zachary Morrow

Publications and source records attributed to Zachary Morrow.

7 recordsLinked to original sources

arXiv2026

Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators

Wildfires are a major producer of fine particulate matter, impacting human health and the electrical grid. Accurately forecasting smoke impacts over long time scales incorporates fuel treatment strategies, natural fuel succession, and stochastic events like lightning strikes. However, predicting smoke for each fuel distribution with a forward simulation of a coupled fire-atmosphere model is computationally infeasible. Moreover, relatively simple fire models are tractable to run in many long-time scenarios but do not capture smoke transport. We use data-driven multilinear operators to predict a smoke concentration field from knowledge of the time since ignition for two quantities of interest: aerosol optical depth and smoke detection. Our method first computes the principal components of time-since-ignition and smoke concentration fields and then learns a map from powers of the input coefficients to the output coefficients. We apply our learned operator to smoke prediction in the Upper Rio Grande Watershed. After collecting training data, learning the approximation weights on a CPU takes less than 30 seconds, and each forward call takes less than 1 ms. On a proxy for aerosol optical depth, we obtain equal accuracy to Monte Carlo sampling with fewer than half as many coupled model calls. For smoke detection, we obtain an intersection-over-union (IoU) of 0.64 and an area under the receiver operating characteristic curve (AUC) of 0.95 on holdout data. Our method is significantly more accurate than the most similar published smoke classifier, which obtains an IoU and AUC of 0.15 and 0.61, respectively, on a 2015 bushfire in Australia.

cs.LG↗

arXiv2026

Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning

This paper introduces the Kernel Neural Operator (KNO), a provably convergent operator-learning architecture that utilizes compositions of deep kernel-based integral operators for function-space approximation of operators (maps from functions to functions). The KNO decouples the choice of kernel from the numerical integration scheme (quadrature), thereby naturally allowing for operator learning with explicitly-chosen trainable kernels on irregular geometries. On irregular domains, this allows the KNO to utilize domain-specific quadrature rules. To help ameliorate the curse of dimensionality, we also leverage an efficient dimension-wise factorization algorithm on regular domains. More importantly, the ability to explicitly specify kernels also allows the use of highly expressive, non-stationary, neural anisotropic kernels whose parameters are computed by training neural networks. We present universal approximation theorems showing that both the continuous and fully discretized KNO are universal approximators on operator learning problems. Numerical results demonstrate that on existing benchmarks the training and test accuracy of KNOs is closely comparable to or higher than that of popular neural operators while typically using an order of magnitude fewer trainable parameters, with the more expressive kernels proving important to attaining high accuracy. KNOs thus facilitate low-memory, geometrically-flexible, deep operator learning, while retaining the implementation simplicity and transparency of traditional kernel methods from both scientific computing and machine learning.

cs.LG↗

arXiv2025

An Optimal Weighted Least-Squares Method for Operator Learning

We consider the problem of learning an unknown, possibly nonlinear operator between separable Hilbert spaces from supervised data. Inputs are drawn from a prescribed probability measure on the input space, and outputs are (possibly noisy) evaluations of the target operator. We regard admissible operators as square-integrable maps with respect to a fixed approximation measure, and we measure reconstruction error in the corresponding Bochner norm. For a finite-dimensional approximation space $V$ of dimension $N$, we study weighted least squares estimators in $V$ and establish probabilistic stability and accuracy bounds in the Bochner norm. We show that there exist sampling measures and weights - defined via an operator-level Christoffel function - that yield uniformly well-conditioned Gram matrices and near-optimal sample complexity, with a number of training samples $M$ on the order of $N \log N$. We complement the analysis by constructing explicit operator approximation spaces in cases of interest: rank-one linear operators that are dense in the class of bounded linear operators, and rank-one polynomial operators that are dense in the Bochner space under mild assumptions on the approximation measure. For both families we describe implementable procedures for sampling from the associated optimal measures. Finally, we demonstrate the effectiveness of this framework on several benchmark problems, including learning solution operators for the Poisson equation, viscous Burgers' equation, and the incompressible Navier-Stokes equations.

math.NA↗

arXiv2025

SUPN: Shallow Universal Polynomial Networks

Deep neural networks (DNNs) and Kolmogorov-Arnold networks (KANs) are popular methods for function approximation due to their flexibility and expressivity. However, they typically require a large number of trainable parameters to produce a suitable approximation. Beyond making the resulting network less transparent, overparameterization creates a large optimization space, likely producing local minima in training that have quite different generalization errors. In this case, network initialization can have an outsize impact on the model's out-of-sample accuracy. For these reasons, we propose shallow universal polynomial networks (SUPNs). These networks replace all but the last hidden layer with a single layer of polynomials with learnable coefficients, leveraging the strengths of DNNs and polynomials to achieve sufficient expressivity with far fewer parameters. We prove that SUPNs converge at the same rate as the best polynomial approximation of the same degree, and we derive explicit formulas for quasi-optimal SUPN parameters. We complement theory with an extensive suite of numerical experiments involving SUPNs, DNNs, KANs, and polynomial projection in one, two, and ten dimensions, consisting of over 13,000 trained models. On the target functions we numerically studied, for a given number of trainable parameters, the approximation error and variability are often lower for SUPNs than for DNNs and KANs by an order of magnitude. In our examples, SUPNs even outperform polynomial projection on non-smooth functions.

cs.LG↗

arXiv2025

Toward real-time optimization through model reduction and model discrepancy sensitivities

Optimization problems arise in a range of scenarios, from optimal control to model parameter estimation. In many applications, such as the development of digital twins, it is essential to solve these optimization problems within wall-clock-time limitations. However, this is often unattainable for complex systems, such as those modeled by nonlinear partial differential equations. One strategy for mitigating this issue is to construct a reduced-order model (ROM) that enables more rapid optimization. In particular, the use of nonintrusive ROMs -- those that do not require access to the full-order model at evaluation time -- is popular because they facilitate optimization solutions can be computed within the wall-clock-time requirements. However, the optimization solution will be unreliable if the iterates move outside the ROM training data. This article proposes the use of hyper-differential sensitivity analysis with respect to model discrepancy (HDSA-MD) as a computationally efficient tool to augment ROM-constrained optimization and improve its reliability. The proposed approach consists of two phases: (i) an offline phase where several full-order model evaluations are computed to train the ROM, and (ii) an online phase where a ROM-constrained optimization problem is solved, $N=\mathcal{O}(1)$ full-order model evaluations are computed, and HDSA-MD is used to enhance the optimization solution using the full-order model data. Numerical results are demonstrated for two examples, atmospheric contaminant control and wildfire ignition location estimation, in which a ROM is trained offline using inaccurate atmospheric data. The HDSA-MD update yields a significant improvement in the ROM-constrained optimization solution using only one full-order model evaluation online with corrected atmospheric data.

math.OC↗

arXiv2024

Glyph-Based Uncertainty Visualization and Analysis of Time-Varying Vector Fields

Uncertainty is inherent to most data, including vector field data, yet it is often omitted in visualizations and representations. Effective uncertainty visualization can enhance the understanding and interpretability of vector field data. For instance, in the context of severe weather events such as hurricanes and wildfires, effective uncertainty visualization can provide crucial insights about fire spread or hurricane behavior and aid in resource management and risk mitigation. Glyphs are commonly used for representing vector uncertainty but are often limited to 2D. In this work, we present a glyph-based technique for accurately representing 3D vector uncertainty and a comprehensive framework for visualization, exploration, and analysis using our new glyphs. We employ hurricane and wildfire examples to demonstrate the efficacy of our glyph design and visualization tool in conveying vector field uncertainty.

cs.HC↗

arXiv2022

Adaptive Basis Sets for Practical Quantum Computing

Electronic structure calculations on small systems such as H$_2$, H$_2$O, LiH, and BeH$_2$ with chemical accuracy are still a challenge for the current generation of the noisy intermediate-scale quantum (NISQ) devices. One of the reasons is that due to the device limitations, only minimal basis sets are commonly applied in quantum chemical calculations, which allow one to keep the number of qubits employed in the calculations at minimum. However, the use of minimal basis sets leads to very large errors in the computed molecular energies as well as potential energy surface shapes. One way to increase the accuracy of electronic structure calculations is through the development of small basis sets better suited for quantum computing. In this work, we show that the use of adaptive basis sets, in which exponents and contraction coefficients depend on molecular structure, provide an easy way to dramatically improve the accuracy of quantum chemical calculations without the need to increase the basis set size and thus the number of qubits utilized in quantum circuits. As a proof of principle, we optimize an adaptive minimal basis set for quantum computing calculations on an H$_2$ molecule, in which exponents and contraction coefficients depend on the H-H distance, and apply it to the generation of H$_2$ potential energy surface on IBM-Q quantum devices. The adaptive minimal basis set reaches the accuracy of the double-zeta basis sets, thus allowing one to perform double-zeta quality calculations on quantum devices without the need to utilize twice as many qubits in simulations. This approach can be extended to other molecular systems and larger basis sets in a straightforward manner.

physics.chem-ph↗