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Operator Learning

Heidarinezhad, Sajjad | 2025

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 58428 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Fotouhi, Morteza
  7. Abstract:
  8. Operators play a fundamental role in mathematics, and understanding the relationships between functions has wide-ranging applications in physics, engineering, economics, and finance. Examples of such applications include solving partial differential equations, modeling complex physical systems, generative modeling, inverse problems, and Bayesian inference. According to the universal approximation theorem, neural networks can approximate any continuous function. A lesser-known but powerful extension of this theorem states that even a neural network with a single hidden layer can approximate any continuous nonlinear operator. Consequently, deep neural network architectures can approximate continuous nonlinear operators with high accuracy, even when the available data is relatively limited. The goal of operator learning is to discover the underlying characteristics of a dynamical system or a partial differential equation from data. By providing a dataset consisting of input-output pairs ("system output" ,"system input" ), one can train a neural network to generalize effectively to previously unseen inputs. This allows for robust performance beyond the training set. Moreover, operator learning can overcome limitations of traditional methods, such as the computational challenges of solving high-dimensional differential equations or the instability of generative models that fail to accurately capture probability distributions. This research focuses on learning operators from data and aims to develop accurate and efficient methods for approximating complex operators. The proposed approaches have the potential to address a broad range of problems in physics, engineering, computer science, financial mathematics, and beyond
  9. Keywords:
  10. Deep Learning ; Option Pricing ; Computational Machine Learning ; Financial Mathematics ; Computational Operators

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