| Abstract: |
| This study presents a hybrid computational framework integrating physics-informed neural networks (PINNs) with stabilized finite element methods for simulating haptotaxis-driven cancer invasion dynamics. The governing equations consist of time-dependent, nonlinear, coupled partial differential equations describing cancer cell density, extracellular matrix (ECM) degradation, and matrix-degrading enzyme concentration. In convection-dominated regimes, standard Galerkin finite element methods produce spurious oscillations and nonphysical negative densities. To address this, we employ the streamline-upwind/Petrov--Galerkin (SUPG) formulation augmented with the YZ$\beta$ discontinuity-capturing technique to generate numerically stable reference solutions. A multi-phase adaptive PINN training strategy is then proposed, progressively transitioning from data-dominant learning using SUPG--YZ$\beta$ solutions to physics-informed refinement through selective enforcement of governing equations. The neural network architecture incorporates Fourier feature embeddings and deep residual blocks to capture sharp tumor invasion fronts. Numerical experiments on benchmark haptotaxis models demonstrate that the hybrid approach eliminates nonphysical oscillations while achieving enhanced accuracy compared to standalone stabilized FEM, effectively resolving steep gradients characteristic of invasive tumor boundaries. The framework is implemented using FEniCS and PyTorch with GPU acceleration, providing a flexible tool for mathematical oncology applications. |
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