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Hybrid simulators for cosmology
Carol Cuesta-Lazaro and Chirag Modi
IAIFI Fellow - AstroAI
![](https://simonsfoundation.imgix.net/wp-content/uploads/2021/06/23162823/Learning-the-Universe-Summary-Figure.png?auto=format&q=90)
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Learning The Universe (LtU)
![](https://abacussummit.readthedocs.io/en/latest/_images/abacussummit_dpi192_portrait.png)
![](https://www.tng-project.org/static/tng/images/figure_tng50_example_galaxies_z1z2.jpg)
TNG50
![](https://sjeffreson.github.io/images/galaxy-fig-insets-smaller.png)
Image credit: Sarah Jeffreson
Image credit: Lehman Garrison, AbacusSummit
\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} a } = \frac{1}{a^3 E(a)}\mathbf{v}
\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} a } = \frac{1}{a^2 E(a)}\mathbf{F}(\mathbf{x},a)
Physics Simulator
x_{\rm Initial}
x_{\rm Final}
Neural Network
x_{\rm Initial}
x_{\rm Final}
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Hybrid Model
\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} a } = \frac{1}{a^3 E(a)}\mathbf{v}
\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} a } = \frac{1}{a^2 E(a)}\mathbf{F}(\mathbf{x},a) +
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x_{\rm Initial}
x_{\rm Final}
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10853760/densities-1.png)
Nbody
Slow
Non-differentiable
Particle mesh
Accurate
Fast
Differentiable
Missing small scales
Nbodyify
Fast
Differentiable
Accurate
![](https://i.pinimg.com/474x/32/09/a7/3209a7c6b90087e89e6aaf9cd7ae325c.jpg)
![](https://cdn.memes.com/up/98946491600539583/i/1605563047040.jpg)
![](https://i.pinimg.com/474x/32/09/a7/3209a7c6b90087e89e6aaf9cd7ae325c.jpg)
"Nbodyify: adaptive mesh corrections for PM simulations" Carolina Cuesta-Lazaro, Chirag Modi (in prep)
\frac{\mathrm{d} \mathbf{x}}{\mathrm{d} a } = \frac{1}{a^3 E(a)}\mathbf{v}
\frac{\mathrm{d} \mathbf{v}}{\mathrm{d} a } = \frac{1}{a^2 E(a)}\mathbf{F}(\mathbf{x},a)
\mathbf{F}_\theta(\mathbf{x},a) = \frac{3 \Omega_m}{2} \nabla \left[\phi^\mathrm{PM}(\mathbf{x}) + \phi^\mathrm{corr}_\theta(\mathbf{x}, a, \phi^\mathrm{PM}, \delta^\mathrm{PM}) \right]
\mathbf{F}(\mathbf{x},a) = \frac{3 \Omega_m}{2} \nabla \phi^\mathrm{PM}(\mathbf{x})
Gravitational evolution ODE
Particle-mesh
Hybrid Simulator
\mathcal{L} = \sum_t \left(x_t^{\rm pred} - x_t^{\rm HR}\right)^2
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10826038/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10826039/pasted-from-clipboard.png)
\delta_\mathrm{LR}
\phi_\mathrm{LR}
Density
Gravitational Potential
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1. CNN
2. Read features at position using attention
h_\theta(\mathbf{x})
\mathbf{F}_\theta(\mathbf{x},a) = f_\theta(h_\theta(\mathbf{x}), a)
3. Compute correction to force
4. Run corrected simulation
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10826217/pasted-from-clipboard.png)
d_{\rm pm}
W_{\rm m} = f_\theta(d_{\rm pm}, h_{\rm m})
h(x) = \sum_m w_{\rm m} h_{\rm m}
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![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10826116/pasted-from-clipboard.png)
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![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10812245/densities-1.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10825999/pasted-from-clipboard.png)
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10825999/pasted-from-clipboard.png)
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Power Spectrum
Cross correlation
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Power Spectrum
Cross correlation
![](https://s3.amazonaws.com/media-p.slid.es/uploads/993552/images/10953160/pasted-from-clipboard.png)
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Improving small scales?
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p(x_{\rm LR}|x_{\rm HR})
Do we need particle-particle interactions?
Turn probabilistic!
deck
By carol cuesta
deck
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