aiqbench analysis
This paper distinguishes SciML optimization from classical ML by highlighting how physics-informed constraints and differential operators create loss landscapes dominated by spectral properties rather than data statistics. Consequently, it argues that standard stochastic methods often fail in this regime, necessitating deterministic or curvature-aware approaches that account for global coupling and stiffness. The work serves as a foundational reference for practitioners and researchers navigating the algorithmic requirements at the intersection of scientific computing and machine learning.
Introduction to optimization methods for training SciML models
arXiv:2601.10222v2 Announce Type: replace-cross Abstract: Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In contrast, SciML often involves physics-informed or operator-constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature-aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first- and second-order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics-constrained and data-driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning.
Abstract:Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In contrast, SciML often involves physics-informed or operator-constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature-aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first- and second-order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics-constrained and data-driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning.
Numerical Analysis (math.NA); Artificial Intelligence (cs.AI); Optimization and Control (math.OC)
arXiv:2601.10222 [math.NA]
(or arXiv:2601.10222v2 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2601.10222 Focus to learn more arXiv-issued DOI via DataCite
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