diff --git a/NNIntegrators.ipynb b/NNIntegrators.ipynb index 041ddf8..ac91ade 100644 --- a/NNIntegrators.ipynb +++ b/NNIntegrators.ipynb @@ -8,7 +8,7 @@ "Let $q(t)=(x(t),p(t))\\in\\mathbb{R}^{2d}$ be the state vector. Two methods are implemented to advance this state vector in time:\n", "\n", "### Multistep neural network\n", - "The `MultistepNNIntegrator` class in [nn_integrator.py](./src/nn_integrator.py) is used to implement a neural network based integrator with a $S$-step method. This is integrating a given $d$-dimensional system $\\frac{dq(t)}{dt}=\\mathcal{N}(q(t))$. The underlying neural model implements the mapping\n", + "The `MultistepNNIntegrator` class in [nn_integrator.py](./src/nn_integrators/nn_integrator.py) is used to implement a neural network based integrator with a $S$-step method. This is integrating a given $d$-dimensional system $\\frac{dq(t)}{dt}=\\mathcal{N}(q(t))$. The underlying neural model implements the mapping\n", "\n", "$$\n", "q^{(t-(S-1)\\Delta t)},\\dots,q^{(t-\\Delta t)},q^{(t)} \\mapsto q^{(t+\\Delta t)}\n", @@ -25,7 +25,7 @@ "* it can be a two-layer LSTM network, followed by a dense layer as in [Kadupitiya et al. (2020)](https://arxiv.org/abs/2004.06493)\n", "\n", "### Hamiltonian neural network integrator\n", - "Alternatively, the `HamiltonianVerletNNIntegrator` in [nn_integrator.py](./src/nn_integrator.py) implements a single-step Stoermer-Verlet method for a Hamiltonian system, following the ideas in [Greydanus et al. (2019)](https://arxiv.org/abs/1906.01563). In this case the update $q^{(t)}\\mapsto q^{(t+\\Delta t)}$ takes the form:\n", + "Alternatively, the `HamiltonianVerletNNIntegrator` in [nn_integrator.py](./src/nn_integrators/nn_integrator.py) implements a single-step Stoermer-Verlet method for a Hamiltonian system, following the ideas in [Greydanus et al. (2019)](https://arxiv.org/abs/1906.01563). In this case the update $q^{(t)}\\mapsto q^{(t+\\Delta t)}$ takes the form:\n", "\n", "$$\n", "\\begin{aligned}\n", @@ -46,7 +46,7 @@ "\n", "![Dynamical systems](figures/DynamicalSystems.svg)\n", "\n", - "The Hamiltonians of the considered dynamical system are written down explicitly in the following sections. For further details, also see the implementation in [dynamical_system.py](./src/dynamical_system.py).\n", + "The Hamiltonians of the considered dynamical system are written down explicitly in the following sections. For further details, also see the implementation in [dynamical_system.py](./src/nn_integrators/dynamical_system.py).\n", "\n", "### Harmonic oscillator\n", "\n", @@ -90,11 +90,6 @@ "H(q,p) = m g L \\left( 2-\\cos(q_0)-\\cos(q_1) \\right) + \\frac{1}{2}k_{\\text{spring}} \\left(|\\boldsymbol{x}_0-\\boldsymbol{x}_1|-d\\right)^2 + \\frac{1}{2 m L^2}\\left(p_0^2 + p_1^2\\right)\n", "$$" ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [] } ], "metadata": {