Advanced Methods

MomentEmu

A general-purpose polynomial emulator based on moment projection, introduced in Zhang (2025). Given simulation data {(θ⁽ⁱ⁾, y⁽ⁱ⁾)}, MomentEmu constructs a moment matrix M_αβ and moment vector ν_α, then solves the linear system M c = ν to obtain closed-form polynomial coefficients — no iterative optimisation needed.

Key properties: millisecond-level evaluation, negligible training cost (seconds, not hours), fully symbolic & differentiable output, forward (θ → y) and inverse (y → θ) modes, auto-differentiation via JAX, PyTorch, and SymPy.

The MomentEmu workflow: standardise data, assemble moment matrices, solve for polynomial coefficients, and select the optimal degree via validation RMSE.

PolyCAMB — CMB Emulation with MomentEmu

Proof-of-concept application of MomentEmu to cosmological parameter estimation using CAMB. Two emulators are provided:

• PolyCAMB-Dℓ — maps six ΛCDM parameters to the CMB power spectra (TT, EE, BB, TE) at sub-percent accuracy over ℓ ≤ 4050. Training on ~47k simulations takes ~9 s; a full spectrum evaluation takes ~1.5 ms.
• PolyCAMB-peak — bidirectional mapping between cosmological parameters and acoustic peak features (locations & amplitudes of the first five peaks). The resulting symbolic expressions recover the known analytic dependences of peak-height ratios on Ω_bh², Ω_ch², and n_s.

MCPost

Comprehensive Monte Carlo post-analysis package. Provides global sensitivity analysis (mutual information, distance correlation, permutation importance, GP surrogates with ARD, Sobol’ indices) and Monte Carlo / quasi-Monte Carlo integration with importance sampling. Modular, typed, and extensible via a registry-based plugin system.

KME

Numerical solver for the Fokker-Planck equation using the Kramers-Moyal expansion. Tracks the time evolution of probability density functions for stochastic processes, with built-in visualisation of PDF evolution at different time slices. Supports user-defined drift and diffusion coefficients.