PINN-GM: Neural Networks for RTE

Updated 4 November 2025

PINN-GM is a framework that embeds radiative transfer equations into neural network loss functions to ensure conservation and symmetry.
It utilizes advanced techniques like micro-macro decomposition and asymptotic-preserving losses to manage high-dimensional, multiscale problems.
The approach is applied across astrophysics, thermal engineering, and more, enabling robust, scalable solutions for forward, inverse, and coupled simulations.

Physics-Informed Neural Networks (PINNs) for Simulating Radiative Transfer, especially as exemplified by the PINN-GM family, constitute a paradigm in which neural networks are directly constrained by the governing physical laws of radiative transfer. This integration ensures that solutions respect key conservation and symmetry properties, even when solving forward, inverse, or coupled problems that are high-dimensional, multiscale, or otherwise computationally prohibitive for legacy methods.

1. Foundations and Motivation

The radiative transfer equation (RTE) underpins simulations across astrophysics, remote sensing, thermal engineering, biomedical optics, and materials science. The equation takes the general form: $\omega \cdot \nabla_x u + k u + \sigma \left( u - \int_S \Phi(\omega, \omega')u(\omega') d\omega' \right) = f,$ where the angular redistribution kernel $\Phi$ , absorption coefficient $k$ , scattering coefficient $\sigma$ , and source $f$ may vary spatially, spectrally, or temporally. The curse of dimensionality and multiscale structure (e.g., optically thick/thin limits, stiff source terms) lead to severe numerical bottlenecks for grids or mesh-based methods.

Physics-informed neural networks address these limitations by embedding the RTE—and when needed, its related conservation laws and auxiliary conditions—directly as soft constraints into the network's loss function. This design enforces consistency with the true PDE solution at arbitrary collocation points, using automatic differentiation for derivatives and network-based parameterization to represent the high-dimensional state space. PINN-GM is used here as an Editor's term to refer to advanced PINN methods tailored for gray-medium radiative transfer and their multiscale extensions.

2. Neural Network Architecture and Loss Construction

The architectural design of PINN-GM solvers is informed by the specific form and domain of the RTE. The general structure includes:

A deep, fully connected feed-forward neural network to learn the mapping:

$(t, x, \omega, \nu) \mapsto u_\theta(t, x, \omega, \nu)$

for state variables of interest.

Collocation points for enforcing the PDEs are picked via low-discrepancy sampling (e.g., Sobol sequence, Gauss-Legendre quadrature) to mitigate sampling errors in high dimensions (Mishra et al., 2020).
The composite loss function involves multiple terms:
- PDE residuals (interior, boundary, initial),
- Auxiliary constraints (conservation laws, additional ODE/PDEs for coupled systems or auxiliary variables),
- Data mismatch for inverse problems.

Golden examples include:

The "vanilla" PINN loss (Mishra et al., 2020):

$J(\theta) = \sum_{\rm interior} |{\rm PDE\ residual}|^2 + \sum_{\rm boundary} |{\rm BC\ residual}|^2 + \sum_{\rm initial} |{\rm IC\ residual}|^2$

Auxiliary losses to enforce additional physical constraints, e.g., macro-micro decompositions or Legendre auxiliary variables for handling phase functions and anisotropic scattering (Riganti et al., 2023, Li et al., 2022).
Adaptive exponential weighting in the loss to ensure asymptotic-preserving (AP) behavior across kinetic and diffusive regimes (Li et al., 4 Mar 2024).

Automatic differentiation enables exact evaluation of all required derivatives in the loss, avoiding explicit numerical differencing.

3. Advanced PINN-GM Strategies: Multiscale and Asymptotic-Preserving Design

Standard PINNs face stability and accuracy issues in multiscale regimes, particularly when the Knudsen number ( $\varepsilon$ ) is small and the solution transitions to a diffusion-like limit. Innovations in PINN-GM include:

Micro-macro decomposition: Separating the solution into equilibrium (macro) and nonequilibrium (micro) parts, e.g., $f(x, v) = \rho(x) + \varepsilon g(x, v)$ $f (x, v) = ρ (x) + ε g (x, v)$ (Lu et al., 2021, Li et al., 2022).
- The governing equations are recast for $(\rho, g)$ , with a PINN (or coupled PINNs) enforcing the residuals for both.
- This decomposition is essential for uniform error control and stability—plain PINNs may otherwise fail catastrophically as shown for small $\varepsilon$ (Lu et al., 2021).
Asymptotic-preserving loss functions: Incorporating hybrid losses that emphasize the appropriate limiting behavior (transport or diffusion) using regime-adaptive weights (Li et al., 4 Mar 2024, Li et al., 2022, Xie et al., 20 May 2025).
Auxiliary and coupled networks: Introducing extra outputs for integrals or expansion coefficients (e.g., Legendre expansion), or coupling with networks for temperature/electron density in gray radiative transfer equations (Riganti et al., 2023, Xie et al., 20 May 2025).

APNN and MD-APNN variants guarantee accurate convergence both in kinetic and diffusion regimes. RT-APNN further addresses training efficiency, long-term evolution via pre-training/curriculum learning, and sharp interface tracking using MCMC adaptive sampling (Xie et al., 20 May 2025).

4. Simulation of Forward, Inverse, and Coupled Problems

PINN-GM methods excel in forward modeling, inversion, and coupled-physics configurations:

Forward solution of RTEs: Accurate resolution of radiative fields in slab, spherical, or general geometry—including isotropic, Rayleigh, and highly anisotropic scattering—by enforcing the full integro-differential PDE (Mishra et al., 2020, Riganti et al., 2023, Chen et al., 2022).
Inverse problems: Efficient simultaneous recovery of physical parameters (e.g., single scattering albedo, opacity, phase function coefficients) from sparse boundary or distributed data (Riganti et al., 2023, Mishra et al., 2020).
Coupled systems: Handling joint radiative and conductive (or hydrodynamic) evolution, with networks for both physical fields and joint optimization (Keller, 8 May 2025, Riganti et al., 2023).
High-dimensional and stochastic inputs: Demonstrated robustness on inverse problems with limited/noisy boundary data, high-dimensional parameterizations, or even random refractive indices (Baty, 27 Feb 2025, Li et al., 4 Mar 2024, Murari et al., 19 Dec 2024).

Auxiliary PINN architectures (APINN, MA-APNN, etc.) avoid explicit quadrature or mesh-based discretization of integral terms, which is particularly advantageous for high-dimensional transport.

5. Performance, Error Analysis, and Comparative Evaluation

PINN-GM frameworks have been evaluated on canonical and challenging problems:

Accuracy: Subpercent relative errors in standard test-cases, up to order-of-magnitude improvements over classical mesh-based, mesh-free, and even data-driven neural net methods in multiscale RTEs (Lu et al., 2021, Chen et al., 2022, Xie et al., 20 May 2025).
Generalization error: Theoretical $L^2$ error bounds scale with training residuals and the reciprocal of the number of collocation/quadrature points, with no exponential dependence on problem dimension—the curse of dimensionality is mitigated (Mishra et al., 2020, Murari et al., 19 Dec 2024).
Memory and scalability: Unlike grid-based solvers, PINNs' memory grows linearly with network size, not $O(N_x N_\omega N_\nu)$ ; APNN/MA-APNN/RT-APNN designs further reduce parameter counts by architectural coupling (Xie et al., 20 May 2025, Li et al., 4 Mar 2024).
Computational cost: PINNs achieve solutions in minutes to hours for high-dimensional problems where traditional methods may be infeasible.

Numerical benchmarks include the simulation of the non-linear Marshak wave (RT-APNN), gray RTEs in time-dependent and multidimensional settings, slab and spherical geometries, and both stationary and time-dependent transfer with validation against tabulated solutions, Monte Carlo solvers, and analytical references.

6. Physical Consistency, Limitations, and Extensions

The physicality of the solution is strongly enforced:

Conservation laws (mass, energy) are included as explicit or penalty terms in the loss.
Boundary layers and singular phenomena are captured via boundary-layer correctors derived from half-space (Milne) problems (Lu et al., 2021).
The approach is mesh-free, allowing for arbitrary geometries and flexible enforcement of data (including interior or partial information).

Noted limitations include:

For highly non-linear or nonstationary problems, training can become unstable—sparse labeled data or curriculum approaches (MD-APNN, pre-training in RT-APNN) alleviate this (Li et al., 2022, Xie et al., 20 May 2025).
The current methodology primarily applies to LTE or gray-medium problems; extensions to NLTE, polarization, and multidimensional anisotropic scattering are active areas.
In extremely optically thick or thin regimes, adaptive normalization/scaled loss terms are required to avoid trivial solutions (Dahlbüdding et al., 31 Jul 2024).
Model complexity and training time rise with increasing number of output fields or sharp solutions.

Advancements such as APNN, RT-APNN, and auxiliary representations explicitly address several of these challenges, including scalability and long-time integration.

7. Applications and Outlook

PINN-GM methodologies are actively advancing state-of-the-art in:

Astrophysics (supernova, stellar, and exoplanet atmospheric RTEs; reionization; magneto-radiative hydrodynamics) (Chen et al., 2022, Baty, 27 Feb 2025, Keller, 8 May 2025, Korber et al., 2022).
Engineering (nuclear reactor RTEs, high-dimensional radiative heat transfer, materials metrology, semiconductor device modeling) (Riganti et al., 2023, Li et al., 2022).
Biomedical imaging and thermal transport in participating media (Riganti et al., 2023, Murari et al., 19 Dec 2024).

Robust performance in forward, inverse, and mixed-data configurations supports integration into scientific workflows where uncertainty quantification, interpretability, and physical consistency are paramount.

Summary Table: Core PINN-GM Design Strategies and Capabilities

Key Technique	Purpose/Benefit	Cited Example
Macro-micro Decomposition	Uniform stability across regimes (kinetic/diffusive)	(Lu et al., 2021, Li et al., 2022)
AP Loss (MA-APNN, RT-APNN)	Asymptotic consistency, efficiency in multiscale problems	(Li et al., 4 Mar 2024, Xie et al., 20 May 2025)
Auxiliary Outputs	Direct representation of integral and expansion terms	(Riganti et al., 2023)
Adaptive Sampling	Increased efficiency, accuracy for sharp interfaces/waves	(Xie et al., 20 May 2025)
Conservation Constraints	Ensures global physical relationships	(Murari et al., 19 Dec 2024)
Data integration (MD-APNN)	Improved stability/accuracy with sparse labels	(Li et al., 2022)

This synthesis is grounded in recent results demonstrating that advanced PINN-GM architectures deliver accurate, scalable, robust, and physically consistent solutions for radiative transfer in gray and general media, overtaking both classical numerical and data-driven ML methods in contemporary computational science (Mishra et al., 2020, Lu et al., 2021, Li et al., 2022, Riganti et al., 2023, Li et al., 4 Mar 2024, Murari et al., 19 Dec 2024, Xie et al., 20 May 2025).