Deep Learning Approaches to Surrogates for Solving the Diffusion Equation for Mechanistic Real-World Simulations

Area of Science:

• Computational modeling
• Scientific computing
• Machine learning applications

Background:

• Numerical solutions of partial differential equations (PDEs) are computationally intensive, limiting complex spatiotemporal dynamic models.
• Accurate simulations of diffusing species, like oxygen in tissues, are crucial for medical applications such as diabetic retinopathy treatment and fMRI interpretation.
• Current methods for calculating quasi-steady-state solutions for fast-diffusing species are particularly costly.

Purpose of the Study:

• To develop a machine learning surrogate model using a neural network to approximate steady-state diffusion equation solutions.
• To significantly accelerate computationally expensive simulations for dynamic models.
• To enable real-time or near-real-time applications of complex simulations.

Main Methods:

• A Convolutional Neural Network (CNN) was employed to approximate the stationary solution of the diffusion equation.
• The model was trained on a 2D domain with two circular sources and absorbing boundary conditions, simulating oxygen diffusion from blood vessels.
• A roll-back training approach was utilized to enhance convergence by rejecting detrimental stochastic changes.

Main Results:

• The trained neural network surrogate achieved an approximate 1000-fold increase in speed compared to direct numerical calculations for individual simulations.
• The surrogate model provides a viable method for accelerating complex simulations, potentially enabling larger and more detailed models.
• The study discussed various loss functions and accuracy estimators to select optimal networks for specific application requirements.

Conclusions:

• Neural network surrogates offer a significant speed-up for solving diffusion equations, making complex simulations more practical.
• This approach can facilitate advanced applications such as parameter identification and uncertainty quantification, which require numerous simulations.
• Challenges such as overfitting and error analysis were identified, highlighting the need for careful model selection and validation.