Hard-constrained neural networks for modeling nonlinear acoustics

In this computational paper, we model acoustic dynamics in space and time from synthetic sensor data. The tasks are (i) to predict and extrapolate the spatiotemporal dynamics and (ii) to reconstruct the acoustic state from partial observations. To achieve this, we develop acoustic neural networks. These are networks that learn from sensor data, while being constrained by prior knowledge on acoustic and wave physics. The prior knowledge is constrained as a soft constraint, which informs the training, and as a hard constraint (Galerkin neural networks), which constrains parts of the network's architecture as an inductive bias. First, we show that standard feedforward neural networks are unable to extrapolate in time, even in the simplest case of periodic oscillations. This motivates the constraints on the prior knowledge. Second, we constrain the prior knowledge on acoustics in increasingly effective ways by (i) employing periodic activations (periodically activated neural networks), (ii) informing the training of the networks with a penalty term that favors solutions that fulfill the governing equations (soft constrained), (iii) constraining the architecture in a physically motivated solution space (hard constrained), and (iv) a combination of these. Third, we apply the networks on two test cases for two tasks in nonlinear regimes, from periodic to chaotic oscillations. The first test case is a twin experiment, in which the data are produced by a prototypical time-delayed model. In the second test case, the data are generated by a higher-fidelity model with mean-flow effects and a kinematic model for the flame source. We find that (i) constraining the physics in the architecture improves interpolation while requiring smaller network sizes, (ii) extrapolation in time is achieved by periodic activations, and (iii) velocity can be reconstructed accurately from only pressure measurements with a combination of physics-based hard and soft constraints. In acoustics and thermoacoustics, this works opens possibilities for physics-constrained data-driven modeling. Beyond acoustics, this work opens strategies for constraining the physics in the architecture, rather than the training.