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+Differentiable and distributed Particle-Mesh n-body simulations
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Differentiable and distributed Particle-Mesh n-body simulations
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Author
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Wassim Kabalan
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the \(\Lambda\)CDM view of the universe
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+lambda CMD model
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Cosmological Parameters:
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Physical baryon density \(\Omega_b\)
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Physical dark matter density \(\Omega_{cdm}\)
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The age of the universe \(t_0\)
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Scalar spectral index \(n_s\)
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Curvature fluctuation amplitude \(A_s\)
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Reionization optical depth \(\tau\)
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Cosmological probes
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Weak Gravitaional Lensing
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\[ \boxed{e = \gamma + e_i} \] ➢ With \(\gamma\) representing the shear and \(e_i\) representing the intrinsic ellipticity, distributed as \(\mathcal{N}(0, I)\)
+➢ We use the shear as as probe of dark matter distribution
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Traditional cosmological inference
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(Hikage et al. 2018)
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(Hikage et al. 2018)
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➢ Measure the ellipticity \(e = \gamma + e_i\) of all galaxies
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➢ Compute summary statistics based on the 2-point correlation function of the shear field
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➢ Run an MCMC chain to recover the posterior distribution of the cosmological parameters, using an analytical likelihood\[p(\theta | x ) \propto \underbrace{p(x | \theta)}_{\mathrm{likelihood}} \ \underbrace{p(\theta)}_{\mathrm{prior}}\]
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Limitations
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Simple summary statistics works well for Gaussian fields
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The need to compute from theory the likelihood for simple summary statistics
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Beyond 2 point statistics : Forward modeling
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➕ No longer need to compute the likelihood analytically
➖ We need to infer the joint posterior \(p(\theta, z | x)\) before marginalization to get \(p(\theta | x) = \int p(\theta, z | x) \, dz\)
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Possible solutions
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Hamiltonian Monte Carlo
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Variational Inference
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Dimensionality reduction using Fisher Information Matrix
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All require a differentiable fast forward model
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Forward Models in Cosmology
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+Initial Conditions
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+Final Dark Matter
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+Linear field
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+Linear field
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Cosmological simulations WIP
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Fast Particle-Mesh WIP
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Having a full n body simulation is very expensive, so we use a particle mesh simulation
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The idea is to put particles in a grid and evolve the particles in the grid
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you lose some precision on the small scales, but on the large scale it is pretty accurate.
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Numerical cheme
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Cloud in Cell binning
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when we have the forces on the grid point we interpolate the forces on the particles
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+alt text
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Where does LSST fit into this WIP
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Rubin observatory will provide us with a 15 TB of data per night for 10 years
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say that we need to make a cube of big part of the volume of the survey.
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Picture of a 3D force vector field
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High end GPUs have a reached 80 GBs A100 next generation H100
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Problems to consider WIP
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Diffrentability
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Fast GPU
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Distribution of the data
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Multi Host distribution
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JAX
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JAXDECOMP WIP
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example of numpy code and grad
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we can also run multiple devices in a single controlle set up
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80 w=x 80
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Differentiable multi host distributed n body simulation that runs on GPUs
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Related work
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FastPM Poqueres nbody 64^3 pmwd 512^3
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JAXPM WIP
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built on top of JAXDECOMP
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example code of JAXPM
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Three steps
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Generate the linear field
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then LPT simulation
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then the final nbody simulation (use any differential equation solver like diffrax)
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Finally we put the final grid back to the particles
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AutoDiff capabilities WIP
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We can simply differentiate the entire simulation using jax.grad
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We can also interface it with machine learning frameworks like flax for the dark matter halo painting and galaxy painting
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Benchmark WIP
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Animation WIP
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Unique solutions : no, two particles crossing each other yoy can’t tell which one is which
(Hikage et al. 2018)
