Field-Level Forward Modeling

• Field-level forward modeling is a method that models complete spatiotemporal data as direct realizations of underlying physical generative processes.
• It utilizes explicit forward operators to link physical parameters, initial states, and observations, thereby retaining all available information beyond summary statistics.
• This approach finds applications in cosmology, astrophysics, fluid dynamics, and biomedical imaging, significantly enhancing parameter inference and simulation fidelity.

Field-level forward modeling is a methodology in which the full spatiotemporal or pixel-level data are modeled as direct realizations of an underlying physical generative process, bridging initial conditions, model parameters, and data via an explicit forward operator. Unlike approaches based on summary statistics (e.g., power spectra, correlation functions), field-level forward modeling seeks to optimally retain all available information by evaluating the likelihood over the entire data field, consistent with the physics encoded in the forward model and the chosen noise or stochasticity prescription. This approach is central across cosmology, astrophysics, fluid dynamics, geoscience, and biomedical imaging, and is particularly impactful in regimes where nonlinearity, complex selection effects, and instrument response must be incorporated into parameter inference or data simulation.

1. Fundamental Principles and Motivation

Field-level forward modeling rests on the explicit mapping from physical parameters and initial states (e.g., cosmological parameters and primordial fields) to observed data, incorporating all relevant sources of transformation and uncertainty—physical evolution, instrument response, bias, resolution, and noise. The field-level likelihood defines the complete generative model:

$$\mathcal{L}[\mathbf{d} | \theta, \mathbf{s}] \propto \exp\left(-\frac{1}{2} \sum_{i} \frac{\left[ d_i - F_i(\theta, \mathbf{s}) \right]^2 }{ \sigma_i^2 } \right)$$

where $d_i$ are the observed data points (pixels, voxels, time samples), $F_i(\theta, \mathbf{s})$ is the deterministic forward model (possibly including convolutional/integral/field operators), $\theta$ the set of model parameters, $\mathbf{s}$ the latent fields (such as initial density or phase realizations), and $\sigma_i^2$ prescribed or modeled noise variances.

The field-level approach leverages all higher-order statistical modes (beyond correlation functions), accommodates survey complexity, selection functions, and masks via explicit data-model mapping, and enables joint inference over both physical parameters and latent realizations (Cabass et al., 2023, Kostić et al., 2022, Schmittfull et al., 2020, Akitsu et al., 11 Sep 2025).

2. Methodological Frameworks Across Disciplines

1. Cosmological Large-Scale Structure (EFT-based Forward Models): Forward modeling uses Effective Field Theory (EFT) and Lagrangian Perturbation Theory (LPT) to map initial Gaussian density fields to nonlinearly evolved fields, including bias expansions and noise (Stadler et al., 2024, Stadler et al., 2024, Schmittfull et al., 2020, Kostić et al., 2022). Higher-order operators capture gravitational nonlinearity and astrophysical bias; the model is evaluated on a grid with controlled numerical errors.
2. Field-Level Inference in Observational Cosmology: Bayesian pipelines sample over both physical parameters (e.g., amplitude $\sigma_8$, growth rate $f$) and initial conditions, employing Hamiltonian Monte Carlo or block-sampling techniques, with likelihoods directly constructed from residuals between observed and forward-modeled fields (Kostić et al., 2022, Akitsu et al., 11 Sep 2025, Schmittfull et al., 2020). The Gaussian likelihood is well-justified on large scales where the model residuals are nearly Gaussian and uncorrelated; deviations (e.g., Poisson, non-Gaussian noise) require extensions (Akitsu et al., 11 Sep 2025).
3. Astrophysical Scene and Instrument Modeling: In the context of, e.g., supernova photometry in undersampled images, pixel-level forward models represent each image as a sum of convolved sources and backgrounds, optimized with survey-specific instrument models (PSF, detector effects) (Rubin et al., 2021). Parameter estimation proceeds via minimization of the field-level (pixel) residual $\chi^2$.
4. Geophysical and Engineering Applications: Field-level forward models for thermal evolution in complex multi-layer solids use analytical or semi-analytical methods (separation of variables, orthogonal expansions, Green's functions) to efficiently compute temperature fields at high resolution, enabling rapid inversion and uncertainty quantification (Fu et al., 9 Jul 2025).
5. Biomedical Imaging: In EEG, the forward model solves the Poisson equation for the electric potential given detailed spatial distribution of biological conductivity and discretized sources, using H(div)-conforming finite elements to rigorously capture singular dipole currents and their localization (Pursiainen et al., 2016).

3. Implementation Strategies and Numerical Aspects

Field-level forward modeling requires robust and scalable numerical algorithms tailored to both the physical model and the nature of the data:

• Perturbative and Nonlinear Solvers:

Cosmological applications employ LPT (up to third or fourth order), with optimized grid and aliasing control, as in LEFTfield (Stadler et al., 2024, Stadler et al., 2024). For nonlinear or non-perturbative regimes, field-level models are embedded in differentiable simulations (e.g., Particle-Mesh gravity solvers (Valade et al., 3 Feb 2026)).

• Operator Construction and Bias:

Bias expansions are constructed as invariant operators in Lagrangian or Eulerian coordinates, translating to efficient grid-based or convolutional calculations. Transfer functions and counterterms control small-scale modeling uncertainties and grid discretization effects (Obuljen et al., 2022, Belsunce et al., 30 Jun 2025).

• Likelihood Construction and Noise Modeling:

The residual noise field is characterized, with large-scale white noise (shot noise, stochasticity) and, where symmetry allows, leading anisotropic corrections (e.g., $k^2\mu^2$ terms in RSD (Schmittfull et al., 2020)). Correct specification of the likelihood, especially its noise structure, is essential for unbiased inference (Akitsu et al., 11 Sep 2025).

• Sampling and Inference Algorithms:

Hierarchical Bayesian posterior sampling over high-dimensional latent fields and low-dimensional parameters leverages advanced algorithms—Hamiltonian Monte Carlo for latent field blocks, slice or Gibbs sampling for bias/noise/cosmological parameters (Kostić et al., 2022, Valade et al., 3 Feb 2026). Analytical marginalization over linear bias parameters can further enhance efficiency.

4. Applications and Performance Benchmarks

Field-level forward modeling achieves optimal (or nearly so) recovery of physical parameters across scales and contexts:

• Cosmological Parameter Constraints:

Field-level inference matches or slightly outperforms joint power spectrum and bispectrum analyses up to the accuracy permitted by the underlying perturbative modeling, with improvements arising when long–short mode couplings or higher-order stochastic terms are present (Cabass et al., 2023, Akitsu et al., 11 Sep 2025). In contexts such as BAO, field-level approaches can extract additional information by capturing the shape and amplitude of mode-coupling corrections (Schmittfull et al., 2020, Babić et al., 19 May 2025).

• Local Universe Reconstruction:

Bayesian field-level forward modeling of peculiar velocities enables robust inference of the 3D velocity and density fields underlying sparse and noisy radial peculiar velocity datasets, outperforming summary-statistic pipelines in the presence of selection biases (e.g., Malmquist bias) (Graziani et al., 2019, Valade et al., 3 Feb 2026).

• Instrumental and Scene Modeling:

Precision photometry for undersampled space telescopes (e.g., Roman SN survey) is achieved by rigorous forward modeling of the scene at the pixel level, jointly fitting astrophysical and instrumental parameters to deliver millimagnitude precision within calibration requirements (Rubin et al., 2021).

• Astrophysical Plasma Diagnostics:

Field-level forward models for solar coronal seismology and MIT-based magnetic field measurements validate inversion techniques against MHD simulations, quantifying accuracy and systematic effects on field strength and angle recovery at the map level (Yang et al., 15 Jan 2026, Gao et al., 2024, Chen et al., 2021).

5. Model Calibration, Uncertainty, and Forward Map Validation

Correct field-level likelihood–model specification, validation against simulations or experimental data, and robust characterization of uncertainties are required for reliable scientific inference:

• Noise & Likelihood Misspecification:

If the actual noise is Poissonian or density-dependent, but a Gaussian likelihood is assumed, cosmological parameters inferred from the field-level likelihood can be systematically biased (up to O(1σ)) when the non-Gaussian noise carries bispectral or higher-order signal (Akitsu et al., 11 Sep 2025). Incorporation of the correct noise model in the field-level likelihood or summary-statistics templates ensures unbiased inference.

• Numerical Convergence & Model Truncation:

Errors from grid discretization, aliasing, and truncation of perturbative expansion are analytically predictable and controlled by scale cutoffs, grid sizes, and operator order. Best-practice heuristics provide prescriptive grid choices and bias expansion depth for a given perturbative scale, ensuring physical modeling error is dominant over numerical error (Stadler et al., 2024, Stadler et al., 2024).

• Validation with Simulations:

Field-level models are benchmarked against high-resolution N-body, hydrodynamic, and synthetic imaging simulations. Percent-level agreement with simulated fields is achievable over the domain of perturbative validity, with field-level cross-correlations >0.99 for $k < k_{\max}$ (Obuljen et al., 2022, Belsunce et al., 30 Jun 2025, Schmittfull et al., 2020).

6. Domain-Specific Extensions and Recent Innovations

Recent advances have extended field-level forward modeling to new domains and modeling paradigms:

• Latent Neural Diffusion Surrogates:

In geological carbon sequestration, conditional neural field encoders and latent-space diffusion models are trained on ensembles of PDE simulation results, enabling mesh-agnostic, uncertainty-aware forward surrogates that can generate entire spatial–temporal fields at the “field level” orders of magnitude faster than direct simulation, and perform zero-shot Bayesian conditioning on new observations (Feng et al., 17 Aug 2025).

• Instrumental PSF and DOF Correction:

In optical imaging (e.g., background-oriented schlieren), object-space, “cone-ray” forward models account for depth-of-field and finite aperture effects, yielding improved reconstructions versus pinhole models especially at low f-numbers (Molnar et al., 2024).

• Efficient Mock Generation and Forecasting:

Differentiable field-level forward models have enabled the generation of large suites of mocks (e.g., for HI intensity mapping, Lyman-α forest, Ly-α–halo cross-correlation) with controlled fidelity, reducing reliance on full simulation pipelines (Belsunce et al., 30 Jun 2025, Obuljen et al., 2022).

7. Outlook, Limitations, and Future Directions

Field-level forward modeling provides a framework for optimal extraction of information from complex datasets with well-defined physical, instrumental, and stochastic structure. Its key strengths include full information retention, systematic effect accommodation, and robust uncertainty characterization. However, its effectiveness rests on the fidelity of both the physical forward model and the noise/likelihood modeling. Non-Gaussianities, small-scale nonlinearity, and model misspecification become progressively more significant as analysis pushes beyond the domain of perturbative expansion or available computational resources. Future work targets robust, simulation-calibrated non-Gaussian likelihoods, integration with machine-learned field-level surrogates for hard-to-model physics, and hybrid approaches combining the power of field-level modeling with summary-statistic robustness (Cabass et al., 2023, Akitsu et al., 11 Sep 2025, Feng et al., 17 Aug 2025).