Performance of Fourier-based activation function in physics-informed neural networks for patient-specific...

Physics-informed neural networks (PINNs) can be used to inversely model complex physical systems by encoding the governing partial differential equations and training data into the neural network. However, neural networks are known to be biased towards learning less complex functions, called spectral bias. This has important implications in modeling cardiovascular flows, where spatial frequencies can vary substantially across anatomies and pathologies (e.g., aneurysms or stenoses). Recent evidence suggests that Fourier-based activation functions have desirable properties, and can potentially reduce spectral bias; however, the performance and adequacy of such Fourier activation functions have not yet been evaluated in patient-specific cardiovascular flow applications.

The performance of sine activation function was evaluated against tanh and swish activation functions in a 1D advection-diffusion problem, an eccentric 2D stenosis model (Re=5000), and a patient-specific 3D aortic model (Re=823) under pulsatile flow conditions. CFD simulations were performed at high spatio-temporal resolution and data points were extracted for training the neural network. The number of training data points were normalized by L/D. The performance of the PINNs framework was evaluated with increasing number of training data points and across all three activation functions.

Our results demonstrate that sine activation function presents desirable characteristics, such as monotonic reduction in errors, relatively faster convergence, and accurate eigen spectra at higher modes, compared to tanh and swish activation functions. Interestingly, for all activation functions, the domain-averaged errors tended to asymptote at ≈15-20% despite substantial increase in training point density. For 2D eccentric stenosis, errors asymptoted at a sensor point density of 40L/D. For 3D patient-specific aorta, this asymptote was achieved at 180L/D for all three activation functions with an error of ≈15% although sine activation function demonstrated relatively faster convergence.

We have demonstrated that Fourier-based activation functions have higher performance in terms of accuracy and convergence properties for cardiovascular flow applications; however, inherent challenges of neural networks (e.g., spectral bias) can limit the accuracy to ≈15% under physiological, 3D patient-specific blood flow conditions.