Siva Viknesh

Siva Viknesh, Younes Tatari, Chase Christenson et al.

Identifying dynamical systems characterized by nonlinear parameters presents significant challenges in deriving mathematical models that enhance understanding of physics. Traditional methods, such as Sparse Identification of Nonlinear Dynamics (SINDy) and symbolic regression, can extract governing equations from observational data; however, they also come with distinct advantages and disadvantages. This paper introduces a novel method within the SINDy framework, termed ADAM-SINDy, which synthesizes the strengths of established approaches by employing the ADAM optimization algorithm. This facilitates the simultaneous optimization of nonlinear parameters and coefficients associated with nonlinear candidate functions, enabling precise parameter estimation without requiring prior knowledge of nonlinear characteristics such as trigonometric frequencies, exponential bandwidths, or polynomial exponents, thereby addressing a key limitation of SINDy. Through an integrated global optimization, ADAM-SINDy dynamically adjusts all unknown variables in response to data, resulting in an adaptive identification procedure that reduces the sensitivity to the library of candidate functions. The performance of the ADAM-SINDy methodology is demonstrated across a spectrum of dynamical systems, including benchmark coupled nonlinear ordinary differential equations such as oscillators, chaotic fluid flows, reaction kinetics, pharmacokinetics, as well as nonlinear partial differential equations (wildfire transport). The results demonstrate significant improvements in identifying parameterized dynamical systems and underscore the importance of concurrently optimizing all parameters, particularly those characterized by nonlinear parameters. These findings highlight the potential of ADAM-SINDy to extend the applicability of the SINDy framework in addressing more complex challenges in dynamical system identification.

Siva Viknesh, Amirhossein Arzani

Scientific machine learning has enabled the extraction of physical insights from high-dimensional spatiotemporal flow data using linear and nonlinear dimensionality reduction techniques. Despite these advances, achieving interpretability within the latent space remains a challenge. To address this, we propose the DIfferentiable Autoencoding Neural Operator (DIANO), a deterministic autoencoding neural operator framework that constructs physically interpretable latent spaces for both dimensional and geometric reduction, with the provision to enforce differential governing equations directly within the latent space. Built upon neural operators, DIANO compresses high-dimensional input functions into a low-dimensional latent space via spatial coarsening through an encoding neural operator and subsequently reconstructs the original inputs using a decoding neural operator through spatial refinement. We assess DIANO's latent space interpretability and performance in dimensionality reduction against baseline models, including the Convolutional Neural Operator and standard autoencoders. Furthermore, a fully differentiable partial differential equation (PDE) solver is developed and integrated within the latent space, enabling the temporal advancement of both high- and low-fidelity PDEs, thereby embedding physical priors into the latent dynamics. We further investigate various PDE formulations, including the 2D unsteady advection-diffusion and the 3D Pressure-Poisson equation, to examine their influence on shaping the latent flow representations. Benchmark problems considered include flow past a 2D cylinder, flow through a 2D symmetric stenosed artery, and a 3D patient-specific coronary artery. These case studies demonstrate DIANO's capability to solve PDEs within a latent space that facilitates both dimensional and geometrical reduction while allowing latent interpretability.