Port-Hamiltonian Systems: Structure Recognition and Applications

Abstract

In this paper, we continue to consider the problem of recovering the port-Hamiltonian structure for an arbitrary system of differential equations. We complement our previous study on this topic by explaining the choice of machine learning algorithms and discussing some details of their application. We also consider the possibility provided by this approach for a potentially new definition of canonical forms and classification of systems of differential equations.

APPENDIX A: SOME TECHNICAL DETAILS

The data considered in Section 3 were processed by a fully convolutional neural network, which had two convolutional layers (activation layer and dropout layer) and a sample normalization layer. The Adam function was chosen as the optimization function, and the root mean square was chosen as the loss function.

The Adam optimizer adjusted the weights according to the following rule:

$${{w}_{t}} = {{w}_{{t - 1}}} - \eta \frac{{{{{\hat {m}}}_{t}}}}{{\sqrt {{{{\hat {\nu }}}_{t}}} + \epsilon }},$$

where ${{w}_{t}}$ are the weights of the tth layer and ${{\nu }_{t}}$ is the step parameter.

$${{\hat {m}}_{t}} = \frac{{{{m}_{t}}}}{{1 - \beta _{1}^{t}}},$$

$${{\hat {\nu }}_{t}} = \frac{{{{\nu }_{t}}}}{{1 - \beta _{2}^{t}}},$$

$${{m}_{t}} = {{\beta }_{1}}{{m}_{{t - 1}}} + (1 - {{\beta }_{1}}){{g}_{t}},$$

$${{\nu }_{t}} = {{\beta }_{2}}{{\nu }_{{t - 1}}} + (1 - {{\beta }_{2}})g_{t}^{2},$$

where g_t is the gradient on the current mini-sample, β_t are hyperparameters with initial values close to 1, while variables m_t and ν_t are the mean and biased deviations for the gradients of the loss functions.

The number of filters for each convolution layer was 64  128. The number of “trained” parameters was 175 162.