Hard- and soft-constrained thermoacoustic neural networks

In thermoacoustic systems, if the heat release is sufficiently in phase with the acoustic pressure, self-excited oscillations can arise. These oscillations are known as thermoacoustic oscillations, which can have detrimental consequences to gas turbines and rocket engines. A typical nonlinear regime of the thermoacoustic dynamics is a limit cycle, which is characterised by a periodic orbit in the phase space. In this work, we develop and analyse physics-aware neural networks to learn periodic solutions of thermoacoustic systems from data. First, in addition to a data-driven loss, a physical residual penalises solutions that violate the conservation of mass, momentum, and energy. Second, periodicity is imposed by introducing periodic activation functions and a trainable angular frequency in the neural networks. Third, acoustic eigenfunctions are employed as spatial modes, while a jump discontinuity in velocity at the flame location is captured by additional discontinuous modes. We test the algorithm on synthetic data generated from a nonlinear time-delayed model of a Rijke tube, as well as a higher-fidelity model with a kinematic flame. We find that (i) physical constraints significantly improve the predictions from noisy or sparse data, (ii) periodic activation functions outperform conventional activation functions in terms of extrapolation capability and convergence rate, and (iii) spatial features such as boundary conditions and discontinuities can be hard-coded in the neural network with an a-priori selected spatial modes. This work opens up possibilities for the prediction of nonlinear thermoacoustics by combining physical knowledge and data.