PINN-GM: Neural Networks for Radiative Transfer

• PINN-GM is a framework that embeds radiative transfer equations directly into neural network training, ensuring solutions respect physical laws.
• It uses fully-connected feed-forward architectures with tailored loss functions to enforce residual minimization and boundary conditions.
• The approach shows robust performance for both forward and inverse problems, offering computational efficiency over traditional grid-based methods.

Physics-Informed Neural Networks for Simulating Radiative Transfer (PINN-GM) are a class of scientific machine learning algorithms designed to solve radiative transfer equations by integrating the governing physical laws directly into the neural network’s training process. Rather than relying solely on observational or simulation data, PINN-GM architectures minimize the residuals of the radiative transfer PDEs across the domain, providing a robust and efficient tool for both forward and inverse problems in radiative transfer.

1. Problem Statement and Theoretical Framework

The radiative transfer equation models the propagation, absorption, and scattering of radiation (such as light, heat, or particles) through a participating medium. The general form of the equation is:

$$\mathbf{\Omega} \cdot \nabla I(\mathbf{x},\mathbf{\Omega}) + \kappa(\mathbf{x}) I(\mathbf{x},\mathbf{\Omega}) = \int_{4\pi} \sigma_s(\mathbf{x},\mathbf{\Omega}',\mathbf{\Omega}) I(\mathbf{x},\mathbf{\Omega}') d\mathbf{\Omega}' + S(\mathbf{x},\mathbf{\Omega})$$

where:

• $I(\mathbf{x},\mathbf{\Omega})$: radiative intensity at position $\mathbf{x}$ in direction $\mathbf{\Omega}$,
• $\kappa(\mathbf{x})$: absorption coefficient,
• $$\sigma_s(\mathbf{x},\mathbf{\Omega}',\mathbf{\Omega})$$: scattering coefficient,
• $S(\mathbf{x},\mathbf{\Omega})$: source term.

PINN-GM formulates this as a residual optimization problem. The neural network is trained to approximate $I(\mathbf{x}, \mathbf{\Omega})$, and the loss function penalizes deviations from the radiative transfer equation at a set of collocation points across the spatial and angular domains.

2. Network Architecture and Loss Function

The PINN-GM algorithm employs a fully-connected feed-forward neural network. The network inputs typically include spatial coordinates ($\mathbf{x}$), angular variables ($\mathbf{\Omega}$), and possibly physical parameters (e.g., absorption/scattering coefficients). The output is the predicted radiative intensity $I$.

The loss function $L_{\text{PINN}}$ for forward problems is:

$$L_{\text{PINN}} = \frac{1}{N_r} \sum_{j=1}^{N_r} \left| \mathcal{N}\big( I_\theta(\mathbf{x}_j, \mathbf{\Omega}_j) \big) \right|^2 + \frac{1}{N_b} \sum_{j=1}^{N_b} \left| I_\theta(\mathbf{x}_j^b, \mathbf{\Omega}_j^b) - g_j \right|^2$$

where:

• $\mathcal{N}(\cdot)$ denotes the full radiative transfer operator,
• $I_\theta$ is the neural network’s output parameterized by weights $\theta$,
• $N_r$ and $N_b$ are the numbers of residual and boundary collocation points,
• $g_j$ are the prescribed boundary values.

Training proceeds via standard optimizers (e.g., Adam or L-BFGS). Automatic differentiation computes necessary derivatives (spatial and angular) required for the PDE residual.

3. Theoretical Error Estimates and Robustness

The framework provides theoretical error estimates for PINN solutions. Numerical experiments confirm that the PINN-GM algorithm is robust: it consistently finds accurate solutions and demonstrates resilience against issues such as data sparsity or noise. The error analysis demonstrates that, under appropriate regularity and with sufficient network capacity, the expected error in the PINN solution decreases with increased collocation density and network size.

The approach is described as straightforward to implement and generally faster than classic grid-based numerical solutions for problems with smooth solutions or moderate dimensionality.

4. Inverse Problem Formulation

Beyond solving forward (direct) radiative transfer problems, PINN-GM enables inverse analyses wherein unknown parameters or functions (such as spatially varying absorption or scattering coefficients) are inferred. This is achieved by augmenting the set of neural network parameters to include those representing the unknown quantities, and the loss function is modified to additionally penalize discrepancies between predicted and measured data.

For an inverse problem, the loss becomes:

$$L = L_{\text{PINN}} + \lambda_{\text{data}} \cdot L_{\text{data}}$$

where $L_{\text{data}}$ is a mean squared error term comparing PINN predictions to observed data, and $\lambda_{\text{data}}$ is a regularization coefficient.

The method is reported to solve inverse radiative transfer problems efficiently, recovering unknown spatially distributed parameters in a computationally tractable manner.

5. Implementation Workflow

The essential workflow for implementing PINN-GM on radiative transfer problems comprises the following steps:

Step	Description	Output
PDE and Domain Setup	Specify the full radiative transfer equation and geometry.	Formulation of $\mathcal{N}$ and boundaries
NN Architecture	Define inputs ($\mathbf{x}$, $\mathbf{\Omega}$, parameters), architecture, and outputs.	Untrained NN model
Collocation Points	Sample points in the combined spatial-angular domain.	Dataset for residual and boundary terms
Loss Construction	Formulate residual and boundary loss terms, optionally add data term for inverse problems.	Objective function
Training	Use optimizers (Adam, L-BFGS, etc.) with automatic differentiation.	Trained model parameters
Validation & Error	Compute solution at test points, evaluate accuracy against analytical or benchmark numerical solutions.	Accuracy assessment

Subsequent applications may introduce data comparison or parameter optimization loops for inverse scenarios.

6. Practical Advantages and Limitations

The PINN-GM approach offers several practical benefits for radiative transfer:

• Ease of implementation: The method circumvents mesh generation and can be applied directly to problems with complex domains or boundary conditions.
• Robustness: Demonstrates strong performance in forward and inverse problems under data scarcity or noisy scenarios, as the physical law regularization acts as a strong prior.
• Extensibility: Straightforward to adapt for inverse parameter identification with minimal changes to the loss function.
• Computational efficiency: Typically offers faster solution times compared to grid-based methods for moderately sized, smooth problems.

Nevertheless, as with most PINN frameworks, the approach remains constrained by scalability for very high-dimensional angular or spatial domains, and its performance is contingent on effective sampling strategies and sufficient network expressivity.

7. Impact and Future Directions

The PINN-GM paradigm for radiative transfer demonstrates an overview of physics-based constraints and machine learning approximation, enabling solution of both classic and inverse radiative transfer problems. The general methodology applies beyond radiative transfer to a broad class of linear and nonlinear PDE-constrained models.

Potential future developments may include:

• Enhanced sampling strategies for high-dimensional domains.
• Integration with multi-fidelity or data-driven surrogate models.
• Adaptations for time-dependent, multi-physics, or coupled PDE systems.
• Systematic analysis of convergence rates in larger-scale or highly inhomogeneous scenarios.

This approach exemplifies the ongoing integration of scientific computing and machine learning, presenting a practical template for tackling challenging PDE-constraint problems while ensuring physical consistency.