Hybrid Mechanistic, Data-augmented Modeling as a Common Language in Biomathematics
Abstract
The current proposal pursues a hybrid modeling approach, that transcends the recent work in so-called "grey box modeling" to offer a more principled hybrid mathematical modeling approach that at once admits the encoding of salient physics, while also possessing a data-inspired mathematical form; thus leading to a reduced order, surrogate modeling approach spanning disciplinary domains within Biomathematics. The resulting hybrid models are computationally inexpensive to evaluate and offer great flexibility when treating multi-scale biological phenomena, and favorable mathematical properties when applied to inverse problem solutions. An additional strength of the proposed hybrid modeling approach is that it offers a single, unified framework that represents a unique balance of mechanism and data, combined within a single "language" that domain scientists might comfortably pursue and master, due to its portability across contexts and applications. The proposed approach to hybrid mechanistic, data-augmented modeling takes as its point of departure the classical solution to a PDE: the Green s function (a staple of mechanistic modeling). Such Green s function solutions are a special case under the broader Fredholm Theory of integral equations (the Green s function being a first kind example). In Fredholm Theory, a spectral representation of a kernel function is admitted. In the case of many inverse problems, the kernel represents the unknown system in question, and so the eigenfunctions of the spectral decomposition are not known, a priori. However, as it is not unreasonable to imagine that the mechanistic description of the system in question admits a Green s function solution (many such solutions exist for physical systems described by elliptic, parabolic, and hyperbolic PDEs, as well as more complex contexts such as Navier Stokes), we assume the spectral kernel decomposition holds, but empirical eigenfunctions (i.e. POD modes, PCA bases, etc.) are used to form the kernel function; in lieu of the actual (unknown) operator eigenfunctions. In this way, we adopt a promising mechanistic mathematical structure, but also open the door to data: the empirical eigenfunctions are learned from the observed output of our system of interest. Additionally, our data are compressed in a L2 optimal manner, in terms of orthonormal basis vectors (i.e. the empirical eigenfunctions). These vectors may be collected to form orthonormal matrices, which live, as elements, within the compact Stiefel manifold. Powerful theoretical differential geometric results enable us to move smoothly within the nonlinear compact Stieffel manifold, so as to seamlessly uncover suitable bases for general cases, unseen within prior experimental observations; thus generalizing the method. This approach takes the view that most mechanistic descriptions of biological systems may be afforded using linear operators within small "patches" of the parameter space that instantiate hypothesized, unknown Green s functions for the operator. However, outside of these local patches, the proposed approach respects the underlying, global nonlinearity of the system. Other theoretical differential geometric arguments permit the proposed hybrid modeling approach to move, in a principled manner, from patch-to-patch, through the nonlinear manifold, in a way that preserved important mathematical properties and modeling assumptions. It is believed that the proposed approach has broad applicability to many physics contexts, and by extrapolation biological contexts; thus offering a "common language" for the domain scientist. Additionally, the ability to select the data-inspired form of the kernel function, along with its support, offers the domain scientist selectivity in expressing the type of physical behavior that is to be included within the model.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Feb 14, 2019
- Source ID
- W911NF1810351
Entities
People
- Christopher Earls
Organizations
- Army Contracting Command
- Cornell University
- United States Army