Physics-aware learning of nonlinear limit cycles and adjoint limit cycles

Abstract

Thermoacoustic oscillations occur when the heat released by a flame is sufficiently in phase with the acoustic pressure. Under this condition, the linear instability can saturate to a nonlinear self-excited oscillation with a large amplitude. A typical nonlinear regime is a limit cycle, which is characterised by a periodic orbit in the thermoacoustic dynamics. In this paper, we develop a physics-aware data-driven method to predict periodic solutions using forward neural networks. The physics is constrained in two ways. First, the training is informed by a physical residual, which penalises solutions that violate the conservation of mass, momentum, and energy. Second, periodicity is imposed by introducing periodic activation functions in the neural network. We test the algorithm on a nonlinear time-delayed model of a Rijke tube. Adjoint methods offer a cheap and easy way to calculate the gradients with respect to design parameters, hence we extend our study to learning the adjoint variables of the Rijke system, which also settle onto periodic oscillations. We find that (i) periodic solutions of thermoacoustic systems can be accurately learned with this method, (ii) for periodic data, periodic activations outperform conventional activations in terms of prediction capability beyond the training range, and (iii) under the physical constraints, fewer data is sufficient to achieve a good performance. This work opens up possibilities for the prediction of nonlinear thermoacoustics by combining physical knowledge and data.

Current and emerging deep-learning methods for the simulation of fluid dynamics

“…It is also possible to evaluate this solution at later time-points than those employed for fitting the MLP’s parameters; however, it is not possible to guarantee an accurate or stable solution in this case. While in PINNs, the governing equations are weakly enforced by constraints in the loss function, hard constraints (beyond boundary conditions) enforced in the network architecture are becoming an interesting area of research [107,201,202]. Physics-driven neural solvers have only been applied to two-dimensional laminar flows in geometrically simple domains to date.…”

A spectrum of physics-informed Gaussian processes for regression in engineering

“…Figure 2 very loosely maps some of the available methods that exist in the modeling community onto the knowledge from physics and data axes from the original figure and the black to white spectrum (of course it must be noted here that method location will change according to the specific model type and how it is applied—sometimes significantly so). At the lighter end of the scale, surrogate models are a continually growing area of interest (Kennedy and O’Hagan, 2001; Queipo et al, 2005; Bhosekar and Ierapetritou, 2018; Ozan and Magri, 2022) where emulation of an expensive-to-run physical model is needed. The emerging field of probabilistic numerics (Cockayne et al, 2019; Hennig et al, 2022) attempts to account for uncertainty within numerical modeling schemes and overlaps with communities specifically looking at bias correction and residual modeling (Arendt et al, 2012; Brynjarsdottir and Hagan, 2014; Gardner et al, 2021), where a data-driven component is used to account for error in a physical model in the first case, or behaviors not captured by a physical model in the latter (most often achieved by summing contributions from both elements).…”