Physics Informed Machine Learning Models for PDEs with Applications to Laser Bioeffects

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Seman, M. (2023). Physics Informed Machine Learning Models for PDEs with Applications to Laser Bioeffects. Unc Charlotte Electronic Theses And Dissertations.
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Abstract

Recently, advances in the efficiency of gradient calculation algorithms have sparked anexpansion of neural network applications, particularly in the field of approximating partial differential equation (PDE) solutions. Among the most novel and effective approaches is the Physics-Informed Neural Network (PINN) which enforces the PDE residual onto the network’s loss function by sampling the gradients of the network with respect to its inputs. This allows PINNs to learn PDE solutions through data-driven discovery without requiring input-output training pairs. In addition to PINNs this work also explores operator networks that build off of the lesser-known universal operator approximation theorem to learn the differential operators of PDEs as a nonlinear mapping from inputs to the solution. These operator networks are able to learn efficiently from relatively small datasets and accurately predict solutions for untrained instances of the target PDE. Implementations of deep operator networks (DeepONets), Fourier neural operators (FNO), and PINNs are used to learn solutions to the heat diffusion PDE for short pulse laser interactions incident on multi-layer skin and ocular tissue models. The high-frequency components inherent to the heat diffusion solution within these models provide the opportunity to examine the spectral bias of neural networks to learn the low-frequency components of the solution, which is theoretically supported in Neural Tangent Kernel (NTK) theory. Fourier feature embedding and FNO, which learn parameters directly in Fourier space, overcome this spectral bias by shifting eigenvalues for the network’s NTK, demonstrating significant reductions in network convergence times for high-frequency solution components.

Details
Author

Seman, Matthew

Title

Physics Informed Machine Learning Models for PDEs with Applications to Laser Bioeffects

Physical Description

1 online resource (59 pages) : PDF

Date

2023

Degree Granting Institution

University of North Carolina at Charlotte

Abstract

Recently, advances in the efficiency of gradient calculation algorithms have sparked anexpansion of neural network applications, particularly in the field of approximating partial differential equation (PDE) solutions. Among the most novel and effective approaches is the Physics-Informed Neural Network (PINN) which enforces the PDE residual onto the network’s loss function by sampling the gradients of the network with respect to its inputs. This allows PINNs to learn PDE solutions through data-driven discovery without requiring input-output training pairs. In addition to PINNs this work also explores operator networks that build off of the lesser-known universal operator approximation theorem to learn the differential operators of PDEs as a nonlinear mapping from inputs to the solution. These operator networks are able to learn efficiently from relatively small datasets and accurately predict solutions for untrained instances of the target PDE. Implementations of deep operator networks (DeepONets), Fourier neural operators (FNO), and PINNs are used to learn solutions to the heat diffusion PDE for short pulse laser interactions incident on multi-layer skin and ocular tissue models. The high-frequency components inherent to the heat diffusion solution within these models provide the opportunity to examine the spectral bias of neural networks to learn the low-frequency components of the solution, which is theoretically supported in Neural Tangent Kernel (NTK) theory. Fourier feature embedding and FNO, which learn parameters directly in Fourier space, overcome this spectral bias by shifting eigenvalues for the network’s NTK, demonstrating significant reductions in network convergence times for high-frequency solution components.

Genre

masters theses

Subjects--Topics

Artificial intelligence
Physics

Degree

M.S.

Keywords

Deeponet
Fourier Feature
Fourier Neural Operator
Neural Tangent Kernel
Pinn
Spectral Bias

Subject Area

Applied Physics

Advisor(s)

Jacob, Donald
Khan, Taufiquar

Committee Members

Jacobs, Donald
Khan, Taufiquar
Gbur, Greg
Thomas, Robert

Degree Note

Thesis (M.S.)--University of North Carolina at Charlotte, 2023.

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Rights Holder Information

Copyright is held by the author unless otherwise indicated.

Identifier

Seman_uncc_0694N_13489

Permalink

http://hdl.handle.net/20.500.13093/etd:3276