Overview
Scientific machine learning is the merger of traditional scientific computing, grounded in mathematical rigor, with machine learning techniques that allow for universal approximation and computational accelerations. While machine learning has been great in large data regimes, making these methods accurately obey physical laws is difficult. By understanding the mathematical implications of replacing components of scientific computing methods, better methods can be created to respect the physics while fitting data.
Technical Formulation
The problems of interest span two regimes of increasing complexity. The first is nonlinear elliptic systems, where the solution is determined by global interactions across the entire domain. Even in the linear setting, operators such as the Laplacian possess unbounded spectra — eigenvalues grow without bound, causing the inverse operator to decay toward zero. This spectral decay makes high-frequency solution components difficult to capture in the forward problem and nearly invisible when inverting for parameters. SciML methods operating in this regime must respect this structure, or risk learning a spectrally biased approximation that fails precisely where clinical accuracy matters most. The second regime adds a time dimension, yielding nonlinear parabolic systems such as reaction-diffusion equations. Here the solution evolves through a sequence of elliptic-type solves coupled across time, introducing temporal dependencies that compound the challenges above. In the setting of growth and remodeling, the material response is path-dependent — the current state alone is insufficient to determine future evolution. ML architectures in this regime must account for solution history explicitly, whether through recurrent structure, attention mechanisms, or augmented state representations, to remain consistent with the underlying physics.
Clinical Application
The clinical application is to accelerate the simulations required for performing many-query scenarios such as simulating surgical procedures by perturbing systems in silico. The SciML methods must be robust with good uncertainty quantification to allow for actual clinical use.
References & Resources
- Neural Operator: Learning Maps Between Function Spaces - Kovachki et al. The theoretical foundation for neural operators.
- Fourier Neural Operator for Parametric Partial Differential Equations - Li et al. Fourier Neural Operator.