Learning the solution of partial differential equations (PDEs) with a neural network (known in the literature as a physics-informed neural network, PINN) is an attractive alternative to traditional solvers due to its elegancy, greater flexibility and the ease of incorporating observed data. However, training PINNs is notoriously difficult in practice. One problem is the existence of multiple simple (but wrong) solutions which are attractive for PINNs when the solution interval is too large. In this paper, we propose to expand the solution interval gradually to make the PINN converge to the correct solution. To find a good schedule for the solution interval expansion, we train an ensemble of PINNs. The idea is that all ensemble members converge to the same solution in the vicinity of observed data (e.g., initial conditions) while they may be pulled towards different wrong solutions farther away from the observations. Therefore, we use the ensemble agreement as the criterion for including new points for computing the loss derived from PDEs. We show experimentally that the proposed method can improve the accuracy of the found solution.
Modeling of conservative systems with neural networks is an area of active research. A popular approach is to use Hamiltonian neural networks (HNNs) which rely on the assumptions that a conservative system is described with Hamilton's equations of motion. Many recent works focus on improving the integration schemes used when training HNNs. In this work, we propose to enhance HNNs with an estimation of a continuous-time trajectory of the modeled system using an additional neural network, called a deep hidden physics model in the literature. We demonstrate that the proposed integration scheme works well for HNNs, especially with low sampling rates, noisy and irregular observations.
We propose a grid-like computational model of tubular reactors. The architecture is inspired by the computations performed by solvers of partial differential equations which describe the dynamics of the chemical process inside a tubular reactor. The proposed model may be entirely based on the known form of the partial differential equations or it may contain generic machine learning components such as multi-layer perceptrons. We show that the proposed model can be trained using limited amounts of data to describe the state of a fixed-bed catalytic reactor. The trained model can reconstruct unmeasured states such as the catalyst activity using the measurements of inlet concentrations and temperatures along the reactor.