Application of Physics-Informed Neural Networks (PINNs) for Reaction-Diffusion PDEs Modeling an Avascular Growing Tumor - Master Thesis

Aug 6, 2026

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This Master’s thesis applies deep learning to reaction-diffusion partial differential equations (PDEs) modeling an avascular growing tumor, using physics-informed neural networks (PINNs). The study builds a diffusion-reaction model capturing tumor–nutrient oxygen interaction and performs parameter inference for diffusion coefficients that differ between patients. To improve convergence, dynamic weights are introduced by assigning adaptive importance to PDE, initial, and boundary-condition terms in the loss function, mitigating gradient imbalances. Results indicate strong performance and potential extension to higher-dimensional domains and complex geometries common in numerical methods.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages73

Reading Time3 h 4 min

AccessFree to download

1 Introduction

2 Artificial Neural Networks

2.1 Artificial Intelligence and Machine Learning

2.2 Training of Artificial Neural Networks

3 Physics Informed Neural Networks

3.1 Scientific Machine Learning

3.2 Example: Burger’s Equation

4 Application for Reaction-Diffusion PDEs modeling an avascular growing tumor

4.1 Physical Model

4.2 Loss Function and Dynamic Weights

5 Conclusions and Future Considerations

National Technical University of Athens Master of Science: Computational Mechanics Fluids Section School of Chemical Engineering Application of Physics-Informed Neural Networks (PINNs) for Reaction-Diffusion PDEs modeling an avascular growing tumor Master Thesis Ilias Katsifis Supervisor: Assistant Professor Mihalis Kavousanakis Athens, 2024 This thesis is dedicated to the countless non-human animals used in laboratory testing and experimentation. It is my hope that advanced computational tools and the ethical use of artificial intelligence will pave the way to a more compassionate world, fostering a shared environment with all sentient beings. Acknowledgements First, I would like to express my deepest gratitude to Assistant Professor Mihalis Kavousanakis for his invaluable guidance throughout my Master’s thesis. His enthusiasm for my topic and willingness to help at every stage made this journey both enriching and memorable. Also, I am sincerely thankful to PhD candidate Ioannis Lampropoulos for his clear explanations of the physics behind the equations in my thesis. His patience and insight were instrumental in deepening my comprehension of complex concepts. Last but not least, I am grateful to my family and friends for their unwavering emotional support throughout this journey. Their encouragement and understanding were my anchors during the challenging times, and their belief in me kept me motivated. Abstract This Master’s thesis explores the application of deep learning techniques for reaction-diffusion partial differential equations (PDEs) modeling an avascular growing tumor, specifically utilizing physics-informed neural networks (PINNs) . PINNs are an innovative and effective approach for solving partial differential equations (PDEs) and conducting parameter inference. The study focuses on a diffusion-reaction model that simulates the interaction between a cancerous tumor and the nutrient oxygen. To enhance the convergence and effectiveness of the neural network, a novel method known as dynamic weights was employed. More specifically a weight was assigned in each term of the loss function which includes the PDEs, the initial conditions and the boundary conditions. This technique adjusts the weights of each term in the loss function to address potential gradient imbalances during training. Additionally, parameter inference was performed for diffusion coefficients, which vary between patients due to the personalized nature of these values. The results were highly satisfactory, indicating the potential for extending this approach to higher dimensions and more complex geometries, which are common challenges in numerical methods. Contents 1 Introduction 6 2 Artificial Neural Networks 7 2.1 Artificial Intelligence and Machine Learning ............ 7 2.1.1 Inspiration from biological neural networks ........ 8 2.1.2 Linear Regression ....................... 9 2.1.3 Gradient Based Optimization ................ 10 2.1.4 Jacobian and Hessian Matrices ............... 13 2.1.5 Second order gradient methods-Newton’s method .... 15 2.1.6 BFGS/LBFGS method ................... 16 2.2 Training of Artificial Neural Networks ............... 19 2.2.1 Deep Feedforward Networks ................. 19 2.2.2 Activation functions ..................... 22 2.2.3 Forward Propagation ..................... 24 2.2.4 Backpropagation ....................... 24 3 Physics Informed Neural Networks 27 3.1 Scientific Machine Learning ..................... 27 3.1.1 Data-driven solutions of PDEs ............... 27 3.1.2 Parameter Inference ..................... 29 3.2 Example: Burger’s Equation ..................... 30 3.2.1 MATLAB: Deep Learning Toolbox ............. 30 3.2.2 Solution of Burger’s Equation ................ 33 3.2.3 Parameter Inference in the Burger’s Equation ....... 36 3.3 The advantages and disadvantages of PINNs compared to classical numerical methods ..............

FAQs

What problem does the thesis address?

The thesis addresses solving reaction-diffusion PDEs that model an avascular growing tumor, including how tumor interacts with nutrient oxygen.

What method is used to solve the PDEs?

Physics-informed neural networks (PINNs) are used to approximate PDE solutions while incorporating the governing equations and constraints into training.

How does dynamic weights improve training and results?

Dynamic weights assign adaptive coefficients to each loss term corresponding to the PDE, initial conditions, and boundary conditions, reducing gradient imbalances and improving convergence.

Aug 6, 2026

Upload by Ivy