Physics-aware learning of thermoacoustic limit cycles

In thermoacoustic systems, if the heat release is sufficiently in phase with the pressure, self-excited oscillations with finite amplitudes can occur. A typical nonlinear regime of the dynamics is a limit cycle, which is a periodic orbit in the phase space. We develop physics-aware neural networks that learn these periodic solutions from data. Added to a data-driven loss, a physical residual penalises solutions that violate the conservation of momentum and energy. We impose periodicity by periodic activation functions and a trainable frequency. We employ acoustic eigenfunctions as spatial modes, while a jump discontinuity in velocity at the flame is captured by discontinuous modes. We test the algorithm on a time-delayed model of a Rijke tube and a higher-order model with a kinematic flame. We find that (i) physics constraints significantly improve the predictions from noisy or sparse data, (ii) periodic activations outperform conventional activations in terms of extrapolation capability, and (iii) boundary conditions and discontinuities can be hard-coded with a-priori selected spatial modes. This work opens up possibilities for the prediction of nonlinear thermoacoustics by combining physical knowledge and data.