Data-driven Discovery of Governing Physical Laws

January 17, 2017

Abstract: 

Ordinary and partial differential equations are widely used throughout the engineering, physical, and biological sciences to describe the physical laws underlying a given system of interest. We implicitly assume that the governing equations are known and justified by first principles, such as conservation of mass or momentum and/or empirical observations. From the Schrödinger equation of quantum mechanics to Maxwell’s equations for electromagnetic propagation, knowledge of the governing laws has allowed transformative technology (e.g., smart phones, internet, lasers, and satellites) to impact society. In modern applications such as neuroscience, epidemiology, and climate science, the governing equations are only partially known and exhibit strongly nonlinear multiscale dynamics that are difficult to model. Scientific computing methods provide an enabling framework for characterizing such systems, and the SIAM community has historically made some of the most important contributions to simulation-based sciences, including extensive developments in finite-difference, finite-element, spectral, and reduced-order modeling methods.

Kepler was an early big data scientist. As an assistant to Tycho Brahe, he had access to the best and most well-guarded astronomical data collected to date. Upon Brahe’s untimely death, Kepler was appointed his successor with the responsibility to complete Brahe’s unfinished work. Over the next eleven years, he laid the foundations for the laws of planetary motion, positing the elliptical nature of planetary orbits (see figure 1 below).

Figure 1.

Using Tycho Brahe’s state-of-the-art data, Johannes Kepler utilized geometrical principles in Tabulae Rudolphinae [8] to discover that planetary orbits were actually ellipses. Figure credit: [8] (left) and Creative Commons (right).

The initial success of these methodologies, including sparse regression and genetic algorithms, suggest that one can integrate many concepts from statistical learning with traditional scientific computing and dynamical systems theory to discover dynamical models from data. This integration of nonlinear dynamics and machine learning opens the door for principled versus heuristic methods for model construction, nonlinear control strategies, and sensor placement techniques. Additionally, these new model identification methods have transformative potential for parameterized systems and multiscale models where first principle derivations have remained intractable, such as neuroscience, epidemiology, and the electrical grid.