Artificial Intelligence Hub

Center for Advanced Mathematical Sciences

I'm a computational scientist, interested in discovering the simple rules that animate our complex universe using artificial — *and sometimes natural *— intelligence. My research is broadly driven by fundamental questions concerning the behavior of complex systems: like granular materials, fluids, brains, and social organisms.

Can we automate the process of scientific discovery? Can we distill observations into physically meaningful and predictive variables? My research group tackles these questions using a combination of physics-informed machine learning techniques with applications to fluid dynamics, granular materials, and nonlinear chaotic systems.

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The world is made of things. And those things interact. Sometimes, we know the rules at one scale. But even when we do, it’s hard to bridge the scales. We develop mathematical and computational techniques for multiscale modeling, with a wide range of applications, from fluid dynamics and granular materials, to social dynamics and neuroscience.

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Musical instruments shape musical expression and the compositions that are being written for them. We build electronic musical instruments that maximize the range of expression by combining soft electronics design, algorithmic composition and machine learning.

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Words and their meanings evolve, morph and reproduce in symbiosis with the culture that defines them. We study the dynamics of words, the graphical relationships between them, and develop visualization methods for effective exploration and learning.

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3 videos | 1 article

This series covers a brief introduction to machine learning and deep learning, with practical demonstrations in Python using packages like Scikit-learn and Tensorflow.

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1 article

Most differential equations that describe the behavior of quantities of interest do not account for uncertainties. The PDF method is a powerful technique that transforms deterministic equations into probabilistic equations. This is tutorial introduces the basics.

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2023 - Proceedings of the Royal Society A

Discovering governing equations from partial measurements with deep delay autoencoders.

— Bakarji, J., Champion, K., Nathan Kutz, J., & Brunton, S. L.

A central challenge in data-driven model discovery is the presence of hidden, or latent, variables that are not directly measured but are dynamically important. Takens' theorem provides conditions for when it is possible to augment these partial measurements with time delayed information, resulting in an attractor that is diffeomorphic to that of the original full-state system. However, the coordinate transformation back to the original attractor is typically unknown, and learning the dynamics in the embedding space has remained an open challenge for decades. Here, we design a custom deep autoencoder network to learn a coordinate transformation from the delay embedded space into a new space where it is possible to represent the dynamics in a sparse, closed form. We demonstrate this approach on the Lorenz, Rössler, and Lotka-Volterra systems, learning dynamics from a single measurement variable. As a challenging example, we learn a Lorenz analogue from a single scalar variable extracted from a video of a chaotic waterwheel experiment. The resulting modeling framework combines deep learning to uncover effective coordinates and the sparse identification of nonlinear dynamics (SINDy) for interpretable modeling. Thus, we show that it is possible to simultaneously learn a closed-form model and the associated coordinate system for partially observed dynamics.

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Dimensionally consistent learning with buckingham pi.

— Bakarji, J., Callaham, J., Brunton, S. L., & Kutz, J. N.

In the absence of governing equations, dimensional analysis is a robust technique for extracting insights and finding symmetries in physical systems. Given measurement variables and parameters, the Buckingham Pi theorem provides a procedure for finding a set of dimensionless groups that spans the solution space, although this set is not unique. We propose an automated approach using the symmetric and self-similar structure of available measurement data to discover the dimensionless groups that best collapse this data to a lower dimensional space according to an optimal fit. We develop three data-driven techniques that use the Buckingham Pi theorem as a constraint: (i) a constrained optimization problem with a non-parametric input-output fitting function, (ii) a deep learning algorithm (BuckiNet) that projects the input parameter space to a lower dimension in the first layer, and (iii) a technique based on sparse identification of nonlinear dynamics (SINDy) to discover dimensionless equations whose coefficients parameterize the dynamics. We explore the accuracy, robustness and computational complexity of these methods as applied to three example problems: a bead on a rotating hoop, a laminar boundary layer, and Rayleigh-Bénard convection.

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2021 - Journal of Computational Physics

Data-driven discovery of coarse-grained equations.

— Bakarji, J., & Tartakovsky, D. M.

Statistical (machine learning) tools for equation discovery require large amounts of data that are typically computer generated rather than experimentally observed. Multiscale modeling and stochastic simulations are two areas where learning on simulated data can lead to such discovery. In both, the data are generated with a reliable but impractical model, e.g., molecular dynamics simulations, while a model on the scale of interest is uncertain, requiring phenomenological constitutive relations and ad-hoc approximations. We replace the human discovery of such models, which typically involves spatial/stochastic averaging or coarse-graining, with a machine-learning strategy based on sparse regression that can be executed in two modes. The first, direct equation-learning, discovers a differential operator from the whole dictionary. The second, constrained equation-learning, discovers only those terms in the differential operator that need to be discovered, i.e., learns closure approximations. We illustrate our approach by learning a deterministic equation that governs the spatiotemporal evolution of the probability density function of a system state whose dynamics are described by a nonlinear partial differential equation with random inputs. A series of examples demonstrates the accuracy, robustness, and limitations of our approach to equation discovery.

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