Definition of the problem

GOAL : approximate the unknown $f$ on $\Omega$

Approximation of a dynamical system

Introduce a parametric model

1️⃣

3️⃣

Constrained Hamiltonian systems

Modelling the vector field on $\mathcal{M}$

On $\mathcal{M}$ the dynamics can be written as

With $W(q,p)\equiv 0$ when $P(q)=I$.

⚠️  On $\mathbb{R}^{2n}\setminus\mathcal{M}$ the vector field extends non-uniquely.

Learning constrained Hamiltonian systems

Measuring the approximation quality

A few words on the choice of the integrator

A case where preserving $\mathcal{M}$ helps

Suppose to have just few unknown elements in the expression of the Hamiltonian

As a consequence, one expects a very accurate approximation.

Example with the spherical pendulum:

Similar results preserving $\mathcal{M}$

Scan for an interactive

version of the plot

Thank you for the attention

Celledoni, E., Leone, A., Murari, D., & Owren, B. (2023). Learning Hamiltonians of constrained mechanical systems. Journal of Computational and Applied Mathematics, 417, 114608.

Example with the double spherical pendulum

Example with $\mathcal{Q}=S^2$

On $\mathcal{M}$ the dynamics can be written as

Using an integrator $\Psi^{\Delta t}_{\theta}$ that preserves $\mathcal{M}$

PROS

CONS

On $\mathcal{M}$ the function $H_{\theta}$ is unique

The approximation error is not influenced by the drift from the manifold.

These methods are more involved, and often implicit.

The loss function can become harder to be optimized.

The simulated trajectories in the training are physically meaningful.

Not in general clear the impact this has on the final result.

Case with $\mathcal{M}$ homogeneous

A manifold $\mathcal{M}$ is homogeneous if there is a Lie group $\mathcal{G}$ that defines a transitive group action $\varphi:\mathcal{G}\times\mathcal{M}\rightarrow\mathcal{M}$.

Case with $\mathcal{M}$ homogeneous

A manifold $\mathcal{M}$ is homogeneous if there is a Lie group $\mathcal{G}$ that defines a transitive group action $\varphi:\mathcal{G}\times\mathcal{M}\rightarrow\mathcal{M}$.

A vector field $f$ on $\mathcal{M}$ can be represented as $f(x) = \varphi_*(\xi(x))(x)$, for a function

$\xi:\mathcal{M}\rightarrow\mathfrak{g}\simeq T_e\mathcal{G}$.

Lie Group Methods are a class of methods exploiting this structure and preserving $\mathcal{M}$. The simplest is Lie Euler:

$y_i^{j+1} = \varphi(\exp(\Delta t \,\xi(y_i^j)),y_i^j)$

Basic idea of a class of Lie group methods

$\mathcal{M}$

$y_i^j$

$y_i^{j+1}=\varphi_g(y_i^j)$

$\mathfrak{g}$

$\xi$

$\exp$

$\mathcal{G}$

$\varphi_g$

$\Psi^{\Delta t}$

$0$

$\Delta t \xi(y_i^j)$

$g=\exp(\Delta t\,\xi(y_i^j))$

$f\in\mathfrak{X}(M)$

$f^L\in\mathfrak{X}(\mathfrak{g})$