Perturbative Forward Models

• Perturbative forward models are methodologies that compute system evolution by expanding equations in powers of a small parameter.
• They leverage analytic operators or kernels to derive effective dynamics in fields such as quantum field theory, cosmology, and generative modeling.
• Applications include constructing effective Hamiltonians, bias expansions, and score estimation, with systematic error control and renormalization techniques.

A perturbative forward model refers to any methodology—across quantum field theory, statistical mechanics, cosmology, control, or quantum simulation—in which the evolution, observables, or effective dynamics of a system are computed by expanding the underlying equations (Hamiltonians, SDEs, or mappings) order-by-order in a small parameter (commonly a coupling, noise strength, or interaction amplitude). The “forward” aspect indicates that one starts from initial data (bare states, linear perturbations, etc.) and computes evolved quantities via explicit, perturbatively constructed operators or kernels, often leveraging analytic or semi-analytic expressions. These approaches are widely utilized in deriving effective Hamiltonians in QFT, bias expansions for large-scale structure, score estimation in generative modeling, and field-level inference in cosmology.

1. General Framework of Perturbative Forward Models

Perturbative forward models are designed by expanding solutions to evolution equations in powers of a small parameter, transforming either operators or states. In relativistic quantum field theory, the Renormalization Group Procedure for Effective Particles (RGPEP) exemplifies this, where an operator evolution equation generates effective Hamiltonians in a basis of extended particles via the mapping $q_0 \to q_t$ with $t=s^4$ (Glazek, 2012). The series

$$\mathcal{H}_t = \mathcal{H}_f + g \mathcal{H}_{t1} + g^2 \mathcal{H}_{t2} + g^3 \mathcal{H}_{t3} + g^4 \mathcal{H}_{t4} + \ldots$$

embodies the approach; each term is recursively constructed via nested commutators and form factors that enforce “narrowness”—i.e., interactions are band-diagonal in invariant mass.

In cosmology and large-scale structure, forward modeling uses Lagrangian Perturbation Theory (LPT) or Eulerian expansions, evolving the initial Gaussian density field via displacements $x(q,\tau) = q + s(q,\tau)$ with $s = \sum_n s^{(n)}$ (Schmidt, 2020, Stadler et al., 17 Sep 2024). Bias expansions treat tracer (galaxy/halo) densities in powers and derivatives of the matter fields, and Effective Field Theory (EFT) is applied to systematically absorb unknown subgrid physics, ensuring theoretical control.

Score-based generative modeling introduces yet another frontier for perturbative forward models, where the Kolmogorov operator’s eigenbasis is used for analytic score estimation, circumventing optimization and time-dependent sample generation (Khoo et al., 29 May 2025).

2. Order-By-Order Construction and Mathematical Formalism

At the heart of the perturbative forward model is the recursive construction of solutions. In RGPEP, for example, the $n$th-order effective operator coefficients $\mathcal{H}_{tn}$ satisfy:

• First order: $(\mathcal{H}_{t1})_{ab}^\prime = 0$
• Second order: $$(\mathcal{H}_{t2})_{ab}^\prime = \sum_x A_{t,axb} \mathcal{H}_{t1, ax} \mathcal{H}_{t1, xb}$$
• Third and fourth order: increased combinatorial complexity with nested summations and integrals over intermediate configurations.

Fourth-order terms, e.g.,

$$\mathcal{H}_{t4, ab} = \mathcal{H}_{04, ab} + \sum_x B_{t, axb}^{(123,0)} \mathcal{H}_{04, axb} + \sum_{x,y} [...] \mathcal{H}_{04, axyb} + \ldots$$

compactly encode multi-loop corrections analogous to box diagrams in scalar QFTs, instantaneous interactions, and self-energies.

In cosmological forward models, the evolved tracer field is expressed as

$$\delta_g(k) = \int d^3q \left[1 + b_1 \delta_1(q) + \frac{1}{2} b_2 (\delta_1^2(q) - \sigma^2) + b_{\mathcal{G}_2} \mathcal{G}_2(q) + ...\right] e^{\mathrm{i} k \cdot (q + \Psi_1(q))}$$

with mappings into redshift space incorporating velocity terms and higher-order displacements (Schmidt, 2020, Stadler et al., 7 Nov 2024).

In generative modeling, the score function is expanded in the eigenbasis $\{f_k(x)\}$ of the backward Kolmogorov operator, $$s^{(i)}(t,x) = \sum_k c_k^{(i)}(t) f_k(x) - \beta \partial_{x_i} V(x)$$, with coefficients determined by solving a linear system derived from perturbation-theoretic cost functionals (Khoo et al., 29 May 2025).

3. Renormalization, Form Factors, and Systematic Error Control

Perturbative forward models incorporate renormalization either via explicit counterterms (as in RGPEP), form-factor suppression (e.g., $f_{t,ab} = \exp(-t (ab)^2)$ that narrows interactions in the RG flow), or the bias and noise expansions in EFT of LSS. By integrating out uncontrolled modes and parametrizing residuals—e.g., via analytic functions of wavenumber $k$ in the likelihood’s variance—perturbative control is maintained, and theoretical uncertainties become quantifiable nuisance parameters (Schmidt et al., 2018, Stadler et al., 17 Sep 2024).

Numerical implementations invoke best-practice recommendations for grid sizes and filtering:

• Aliasing errors scale as $k^6$, cross-correlations as $k^3 P(k)$, and are suppressed by using, e.g., $N_{G,\mathrm{Eul}} \geq (3/2) N_{G,\mathrm{in}}$ (“3/2-rule”).
• Truncation errors in high-order LPT are controlled by matching grid resolution to the cutoff scale and desired order in perturbation (Stadler et al., 17 Sep 2024).

4. Applications Across Physical Domains

Quantum Field Theory: Perturbative forward models diagnosed which interaction operators dominate at the effective scale, facilitated systematic renormalization, and allowed construction of band-diagonal Hamiltonians for constituent particles, enabling studies of phenomena like confinement in QCD and heavy-quark bound states (Glazek, 2012, Musakhanov et al., 2020).

Cosmology and Large-Scale Structure: Forward modeling at the field level enables fully Bayesian inference by comparing observed galaxy or halo densities to those predicted from initial conditions and evolved via LPT/EFT. Degeneracies in cosmological parameters (e.g., $b_1$–$\sigma_8$) are broken by including higher-order (quadratic) bias and displacement terms, and simulation-based inference becomes tractable as the forward model can be evaluated in seconds for a full cosmological volume (Schmidt et al., 2018, Cabass et al., 2023, Stadler et al., 17 Sep 2024, Stadler et al., 7 Nov 2024).

Quantum Simulation: In noisy intermediate-scale quantum devices, perturbative quantum simulation allows simulation of many-qubit systems by stochastically applying local unitary operations and reconstructing global dynamics via a perturbative Dyson-series expansion (Sun et al., 2021).

Stochastic Control: In T-PFC, nominal trajectories and linear feedback gains are separated, allowing nearly optimal (third-order) control under uncertainty at greatly reduced computational cost compared to NMPC (Parunandi et al., 2019).

Generative Modeling: Optimization-free diffusion models recast score estimation as a linear algebra problem in a sparse eigenbasis using perturbation theory, enabling accurate high-dimensional density modeling without neural network training or SDE simulation (Khoo et al., 29 May 2025).

5. Likelihood Construction, Inference, and Information Equivalence

Perturbative forward models typically enable analytic calculation of the field-level likelihood, especially in the limit of small noise. For cosmological inference, the likelihood for the tracer field is written as

$$-\ln P(\delta_h|\delta, \{b_O\}, \{\lambda_a\}) = \sum_k^{k_\mathrm{max}} \left[ \frac{1}{2} \ln \sigma^2(k) + \frac{1}{2} \frac{|\delta_h(k) - \delta_{h,\mathrm{det}}(k)|^2}{\sigma^2(k)} \right]$$

where $\sigma^2(k)$ is an analytic function constructed from the bias and EFT noise parameters.

Several studies have rigorously demonstrated that, under perturbative control (cutoff scale $k_\mathrm{max}<k_{NL}$), field-level inference captures the full information content of the power spectrum, bispectrum, and BAO reconstruction. The error bars on cosmological parameters from forward modeling match those from joint summary statistics, and in some cases, field-level methods outperform due to direct marginalization over nuisance/initial conditions (Schmidt et al., 2018, Cabass et al., 2023, Akitsu et al., 11 Sep 2025).

6. Challenges, Limitations, and Future Directions

The accuracy and unbiasedness of the perturbative forward model critically depend on the correct specification of the likelihood and the noise model. If residuals from forward model mismatch (e.g., non-Gaussian noise from shot noise, galaxy formation, or stochastic bias) are not modeled accurately, parameter biases can result (Akitsu et al., 11 Sep 2025). In contrast, summary statistic methods (power spectrum, bispectrum) benefit from averaging over many modes, rendering them less sensitive to non-Gaussian departures. Improving likelihoods to capture realistic noise models is a central challenge as analyses move to smaller scales and higher $k$.

Extension to higher-order perturbation theory, inclusion of more complete sets of bias and derivative operators, robust renormalization schemes (e.g., ratio and hybrid schemes for lattice QCD extraction of parton distributions (Yao et al., 2022)), and hybrid quantum-classical simulation approaches (Sun et al., 2021) are ongoing directions.

7. Summary Table: Key Features Across Physical Domains

Domain	Perturbative Small Parameter	Forward Evolution Target
QFT (RGPEP)	Coupling $g$	Effective Hamiltonian
Cosmology/LSS	Nonlinearity, higher-order bias	Galaxy/halo density field
Quantum simulation (PQS)	Weak subsystem coupling	Time-evolved state, observables
Stochastic control (T-PFC)	Action uncertainty $\epsilon$	Trajectory, cost
Generative modeling (diffusion)	Perturbation from base density	Score function

In all cases, perturbative forward models provide a systematic, order-by-order framework for computing evolved observables from initial data, with explicit control over approximation accuracy, error scaling, and analytic understanding of the information content. They underpin modern approaches in cosmology, QFT, control, generative modeling, and quantum simulation.