Neural Operator Learning for Modeling Cancer Cell Growth Bridging Mechanistic ODEs and Experimental In Vitro Data

Predicting cancer cell proliferation is a fundamental challenge in mathematical oncology, traditionally addressed through mechanistic ordinary differential equations (ODEs). While classical models such as the Gompertz and Logistic growth equations offer high interpretability, they often struggle to assimilate sparse, noisy data from in vitro experiments. This research presents a Scientific Machine Learning (SciML) framework that evolves from traditional numerical modeling to Physics-Informed Neural Networks (PINNs) and Neural Operators. Utilizing classical ODEs as a rigorous benchmark, we first employ PINNs to solve the inverse problem, identifying latent growth and diffusion parameters within a mechanistic context directly from primary experimental data. This data, representing cell counts from in vitro studies at Jarvis Christian University, allows us to validate the PINN's ability to maintain physical consistency where traditional numerical methods may fail due to experimental noise. Expanding beyond instance-specific solvers, we introduce an Operator Learning approach (e.g., DeepONets) to learn the solution operator mapping treatment initial conditions to resulting growth trajectories. We provide a comparative analysis between traditional ODE benchmarks, PINNs, and Neural Operators, specifically addressing the "small-data" regime common in biological laboratory settings. Supported by an NSF CISE-MSI Grant, this interdisciplinary collaboration between the Mathematics and Biology departments demonstrates how embedding classical growth laws into neural architectures enhances generalization across diverse experimental cohorts. This work advances the use of data-driven operators as robust, real-time digital twins for predictive mathematical oncology.