No more adjoints: Calibrating chaotic dynamical systems with weak-form learning

Deterministic chaos poses a major challenge in the data-driven learning of chaotic dynamical systems such as turbulent flows. When learning techniques utilize the mean-squared-error for parameterizing such systems, they suffer from instability due to the use of inaccurate adjoints and fail to capture the macroscopic properties of such systems. This results in a common failure mode - machine learning models for dynamical systems provide short-term accuracy but become unstable and unphysical at a larger time horizon. Consequently, the machine learning techniques provide suboptimal results in many fluid dynamics applications, such as the surrogate modeling of turbulent flows. In this work, we approach the learning of chaotic systems from a different perspective given by an integral formulation of the dynamics. By constructing a least-squares optimization for system identification after convolution with a time-dependent test function, backpropagation of gradients through time is avoided and an adjoint system computation is bypassed completely. We demonstrate that the proposed formulation leads to superior learning of both short-term as well as invariant properties of chaotic dynamical systems at much lower computational costs when compared to classical methods for time-series learning of fluid flows.