Operator Learning Using Random Features for Phase-Field Tumor Models

In mathematical oncology tumor evolution can be described by phase-field equations that model the spatio-temporal dynamics of cell populations together with various biophysical mechanisms. Recent surveys outline multiple families of such models, ranging from the prototypical Cahn-Hilliard equation to multiphase formulations incorporating nonlocal cell adhesion, stochastic fluctuations, and mixed-dimensional couplings to vascular networks. All these partial differential equation systems are highly nonlinear, high-dimensional, and computationally demanding to solve with classical numerical schemes, motivating the development of efficient surrogate models. Supervised operator learning has emerged as a promising data-driven framework for approximating solution operators between infinite-dimensional function spaces. Among theses approaches, the random feature method for operator learning provides a well-balanced approach, combining scalability with rigorous theoretical guarantees. It frames training as a convex quadratic optimization problem, admits convergence and complexity bounds, and can be interpreted as a low-rank approximation of operator-valued kernel ridge regression with close connections to Gaussian process regression. Within this framework, we construct random-features for the Cahn-Hilliard equation as the prototypical model for tumor growth, demonstrating accurate prediction of its solution operator and proving convergence of the method in this setting. Building on these results, we try to extend the framework to the four-species phase-field tumor growth model. The random feature method has seen limited application to partial differential equations, with only a few isolated examples in the literature, such as the Burgers equation or Darcy flow. Our work provides new insights into the method’s capabilities and extends its applicability to more complex partial differential equations. Our numerical experiments illustrate that random-feature operator learning provides a scalable, transferable, and discretization-invariant surrogate for the Cahn-Hilliard equation, offering a promising computational tool for mathematical oncology