Hyperelastic Cardiovascular NN-FE: A Framework for Integrating Nonlinear Finite Element Solvers with Neural Networks...

Accurately predicting tissue mechanics of cardiac valves is crucial for computational biomechanics and surgical planning. Conventional finite elements (FEs) entail high computational costs when modeling real-world applications, whereas purely data-driven approaches lack physical consistency and interpretability. We present a hybrid, physics-informed neural network framework called NN-FE, which combines a feedforward neural network with a nonlinear FE solver to predict the deformation of the tricuspid valve posterior leaflet (TVPL) under quasi-static biaxial loading. For collection of biomechanical data, TVPL tissues were subjected to seven biaxial loading ratios. The tissue response was characterized using a Fung-type hyperelastic model with parameters identified via differential evolution optimization. A six-layer feedforward neural network, comprising five hidden layers and one output layer, predicts pointwise displacements as functions of spatial coordinates, time, and loading ratio. These predictions serve as the trial fields for the FE solver, which enforces equilibrium through Newton-Raphson iterations. The FE-corrected displacement fields provide physics-consistent training targets, enabling the neural network to learn the constitutive behavior without explicit partial differential equation residual terms or empirical loss weighting. Model performance was evaluated using leave-one-ratio-out cross-validation across six case studies. We showed that the proposed framework achieved mean absolute percentage errors (MAPE) less than 15% for displacement predictions and temporal coefficients of determination exceeding 0.85 when predicting out-of-distribution scenarios. Additionally, tissue stretch predictions demonstrated superior accuracy (MAPE < 1.4%), while stress predictions exhibited a MAPE of less than 7.8%. The NN-FE framework represents a computationally efficient surrogate model suitable for real-time applications in cardiovascular biomechanics.