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Fixed broken links
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eikehmueller committed Aug 19, 2022
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11 changes: 3 additions & 8 deletions NNIntegrators.ipynb
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"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",
"$$\n",
"q^{(t-(S-1)\\Delta t)},\\dots,q^{(t-\\Delta t)},q^{(t)} \\mapsto q^{(t+\\Delta t)}\n",
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"* it can be a two-layer LSTM network, followed by a dense layer as in [Kadupitiya et al. (2020)](https://arxiv.org/abs/2004.06493)\n",
"\n",
"### Hamiltonian neural network integrator\n",
"Alternatively, the `HamiltonianVerletNNIntegrator` in [nn_integrator.py](./src/nn_integrator.py) implements a single-step Stoermer-Verlet method for a Hamiltonian system, following the ideas in [Greydanus et al. (2019)](https://arxiv.org/abs/1906.01563). In this case the update $q^{(t)}\\mapsto q^{(t+\\Delta t)}$ takes the form:\n",
"Alternatively, the `HamiltonianVerletNNIntegrator` in [nn_integrator.py](./src/nn_integrators/nn_integrator.py) implements a single-step Stoermer-Verlet method for a Hamiltonian system, following the ideas in [Greydanus et al. (2019)](https://arxiv.org/abs/1906.01563). In this case the update $q^{(t)}\\mapsto q^{(t+\\Delta t)}$ takes the form:\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
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"\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",
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"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": {
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