Physics Informed Neural Networks for Simulating Radiative Transfer

Siddhartha Mishra; Roberto Molinaro

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Physics Informed Neural Networks for Simulating Radiative Transfer

Siddhartha Mishra, Roberto Molinaro

arXiv:2009.13291v3Sep 25, 2020cs.LGstat.ML

TL;DR

Radiative-transfer simulation is difficult because the governing equation is high-dimensional and existing methods can be costly or difficult to use. The paper develops PINN-based forward and inverse algorithms, combining residual-based training with theoretical error estimates and experiments. The reported results show low errors, short training times, and performance comparable to published results, while the inverse extension efficiently estimates unknown coefficients.

Problem
Accurate and efficient radiative-transfer simulation remains challenging because the equation is high-dimensional and existing methods have computational deficiencies.
Method
The paper trains deep neural networks by minimizing a loss containing the radiative-transfer PDE residual and initial and boundary-condition mismatches, with a data-fidelity term for inverse problems.
Results
PINNs produced low errors with small training times, results quantitatively and qualitatively comparable to published results, and error bounds validated by experiments.
Takeaways & Limitations
The paper presents PINN algorithms as general-purpose, simple-to-implement, fast, and accurate tools for forward and selected inverse radiative-transfer problems.
Takeaways & Limitations
The inverse-problem treatment is brief, and careful analysis and practical applications of broader extensions are left for future work.

Abstract

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We propose a novel machine learning algorithm for simulating radiative transfer. Our algorithm is based on physics informed neural networks (PINNs), which are trained by minimizing the residual of the underlying radiative tranfer equations. We present extensive experiments and theoretical error estimates to demonstrate that PINNs provide a very easy to implement, fast, robust and accurate method for simulating radiative transfer. We also present a PINN based algorithm for simulating inverse problems for radiative transfer efficiently.

1 Introduction

Radiative transfer is important across science and engineering, but its high-dimensional equation and heterogeneous optical properties make accurate, efficient simulation difficult. The paper proposes PINN-based forward and inverse algorithms, supported by error analysis and numerical experiments.

Radiative transfer supports applications including astrophysics, climate dynamics, meteorology, nuclear engineering, and medical imaging.
In three spatial dimensions, radiative intensity depends on seven variables, making traditional grid-based methods computationally demanding.
Existing methods trade off convergence speed, implementation complexity, dimensionality, accuracy, or computational cost across different radiative-transfer settings.
The paper aims to provide an accurate, fast, easy-to-use method that handles high dimensionality and optical heterogeneity.
The proposed algorithm uses PINNs to approximate the radiative transfer equation in a general setting.
The authors provide a generalization-error estimate and numerical experiments for the forward problem, and extend PINNs to inverse coefficient recovery.

2 Physics informed neural networks for approximating (1.1)

This section introduces the PINNs algorithm for simulating radiative transfer and begins by describing the underlying radiative-transfer PDE.

The section presents the PINNs algorithm for simulating radiative transfer and first elaborates on the underlying PDE.

2.1 The model.

The model describes radiative intensity in a static medium using an evolution equation with initial and inflow-boundary conditions, while angular moments provide lower-dimensional observables.

Radiative transfer in a static medium is modeled by an evolution equation for radiative intensity u.
The model specifies the initial condition u(0, x, ω, ν) = u0(x, ω, ν).
Boundary conditions are imposed on the inflow boundary, where ω · n(x) < 0.
On the inflow boundary, the radiative intensity is set to the prescribed value ub.
Incident radiation and heat flux are used as physically meaningful angular-moment observables of the high-dimensional radiative intensity.
The steady formulation formally follows by taking c →∞ and dropping the time-derivative term.

2.2 Quadrature rules and Training points

Quadrature approximates the scattering integral and supplies PINN training points, using Gauss–Legendre rules, Sobol sequences, or random points depending on the geometry.

Quadrature is essential for approximating the scattering-kernel integral and supplying PINN training points.
Quadrature approximation specifies points zi and weights wi for integrating an integrable function over a domain.
Gauss–Legendre quadrature rules are used to approximate the scattering-kernel integral.
Training points are constructed from suitable quadrature rules for the relevant domains and variables.
For logically rectangular spatial domains, Sobol sequences provide low-discrepancy points, while complicated geometries use uniformly distributed random points.

2.3 Neural Networks

The paper represents radiative-transfer solutions with fully connected deep neural networks that compose affine maps and scalar nonlinear activations. The network maps radiative-transfer inputs to a scalar output through hidden layers.

Network architecture: PINNs use feedforward neural networks that transform y = (t, x, ω, ν) through successive affine maps and nonlinear activations.The input domain is D = DT × S × Λ, and the network output represents the approximated solution.
Network architecture: The network is written as a composition of affine layers Ck and componentwise activation functions A.The composition includes K layers, with activations applied between affine transformations.
Network architecture: Each affine layer applies Ckzk = Wkzk + bk, with weights and biases determining the layer transformation.The matrices and vectors have dimensions specified by the widths of adjacent layers.
Network architecture: The architecture has an input layer, an output layer, and K − 1 hidden layers, with d1 = 2d + 1 and dK = 1.The output layer is therefore scalar for the modeled radiative-transfer solution.
Illustration: In the illustrated fully connected network, red neurons denote inputs, blue neurons outputs, and yellow neurons hidden units.Connections combine affine maps between layers with scalar nonlinear activation functions within units.

2.4 Training PINNs: Loss functions and optimization

The PINN is trained by minimizing discretized PDE, initial-condition, and boundary-condition residuals over network parameters. Quadrature approximates the residual norms, and gradient-based optimization produces the final approximation.

Objective: The network parameters are trained so that uθ approximates the radiative-transfer solution by minimizing a loss measuring residual mismatch.The optimization may include a regularization term weighted by λreg.
Residuals: The training residuals include the PDE residual, initial-condition residual Rtb = uθ − u0, and spatial-boundary residual Rsb = uθ − ub.These residuals are minimized simultaneously over admissible tuning parameters.
Discretized loss: Because residual norms contain intractable integrals, the loss function approximates them using quadrature points and weights.The balancing hyperparameter λ weights contributions from PDE, initial, and boundary residuals.
Optimization: The algorithm evaluates the network, residuals, loss, and gradients, then applies a non-convex gradient-based optimizer until an approximate local minimum θ∗ is reached.The paper mentions stochastic gradient descent methods such as ADAM and higher-order methods such as LBFGS.
Output: The resulting map u∗ = uθ∗ is the PINN approximation of the radiative-transfer solution.Algorithm 2.1 takes the domain, equation data, quadrature rules, and optimization algorithm as inputs.

2.5 Estimates on the generalization error

The paper derives a generalization-error estimate for the trained PINN on a rescaled computational domain. The bound depends on training and quadrature errors, has logarithmic spatial-dimensional dependence, and has a stated limitation for steady-state problems.

Domain assumptions: The spatial domain is rescaled to D = [0, 1]^d, with rectangular domains mapped there by coordinate transformations.Time, frequency, and angular domains are likewise rescaled in the construction of the training domain.
Error estimate: The analysis targets the generalization error of the trained network u∗ = uθ∗ as an approximation to the radiative-transfer solution.The estimate is expressed in terms of training errors and quadrature-related quantities.
Error estimate: Training errors are computable after optimization, so the bound implies accurate generalization when the PINN is trained well and the constants remain finite.The estimate links post-training residual errors to approximation quality.
Dimensional dependence: The error bound has only logarithmic dependence on spatial dimension, and the paper argues this does not impose a severe restriction for d ≤ 3.The stated decay behavior requires sufficiently many training points and training errors independent of dimension.
Limitation: For steady-state problems obtained formally by c →∞, the time-dependent bound becomes unsuitable because its constant grows exponentially with rescaled time.The paper states that a modified steady-state estimate is derived separately in Appendix B.

3 Numerical Experiments

The experiments evaluate PINNs on one-dimensional monochromatic, three-dimensional monochromatic, and three-dimensional polychromatic radiative-transfer problems. Across these settings, PINNs achieve accurate approximations with low training costs, while the three-dimensional monochromatic case lacks an analytical generalization-error benchmark.

3.2 Monochromatic stationary radiative transfer in one space dimension: The one-dimensional monochromatic experiment uses Sobol training points, Gauss-Legendre quadrature, and ensemble-selected hyperparameters to approximate a potentially discontinuous solution.The experiment uses Nint = 8192, Nsb = 2048, and NS = 10.
3.2 Monochromatic stationary radiative transfer in one space dimension: The PINN closely matches the finite-element solution and accurately approximates the exact discontinuous boundary solution at low computational cost.The finite-element comparison required 20 minutes of CPU training, while the boundary errors are reported as very small in L2-norm.
3.3 Monochromatic stationary radiative transfer in three space dimensions: The three-dimensional monochromatic experiment uses 16,384 interior points, 12,288 boundary points, and 100 quadrature points, obtaining a total training error of 4.4 × 10^-4.The problem is posed in the unit cube with a central source, zero incoming boundary radiation, absorption, and isotropic scattering.
3.3 Monochromatic stationary radiative transfer in three space dimensions: The three-dimensional monochromatic PINN produces incident-radiation results very similar to a discrete ordinate method and trains in approximately 70 minutes on one GPU.The comparison is qualitative because no analytical radiative-intensity solution is available for computing generalization errors.
3.4 Polychromatic stationary radiative transfer in three space dimensions: For steady polychromatic transfer in three dimensions, the PINN achieves approximately 2% relative L2 error for radial flux with one hour of training.The added frequency variable makes this a six-dimensional problem, and no analytical radiative-intensity solution is available for direct generalization-error computation.

4 PINNs for the Inverse problem for radiative transfer

The paper modifies PINNs to solve an inverse radiative-transfer problem by jointly approximating absorption and radiative intensity while enforcing PDE, boundary, and measurement residuals. In a numerical experiment, the method recovered measured radiation closely, absorption within 3%, and radiative intensity with below-1% generalization error in about 100 minutes.

Problem: The inverse task estimates the unknown absorption coefficient and radiative intensity from measurements of incident radiation.
Caveat: The inverse problem is ill-posed because multiple absorption coefficients may produce the same incident radiation, so the method seeks one consistent coefficient.
Method: The modified PINN jointly represents the absorption coefficient k and radiative intensity u with separate neural networks.
Method: The inverse loss combines PDE, data, initial-condition, and boundary residuals, with the data residual comparing computed and measured incident radiation.
Experiment: The experiment used 16,384 interior training points and 4,096 data points, with Sobol points for the interior set and random points for data.
Results: The incident radiation was almost identical to measurements, absorption error was less than 3%, and radiative-intensity generalization error was below 1%.The computation took approximately 100 minutes.

5 Discussion

The discussion presents PINNs as a simple, fast, and accurate approach for high-dimensional radiative-transfer simulation, supported by experiments and dimension-independent error bounds. It also identifies inverse problems and broader extensions as important but incompletely developed directions.

Motivation: Radiative-transfer simulation is challenging because the general problem has seven variables and combines emission, absorption, scattering, and varying optical parameters.
Method: The forward PINN minimizes collocated PDE, initial-condition, and boundary residuals, using Sobol low-discrepancy points to alleviate the curse of dimensionality.
Results: Across experiments from one-dimensional stationary to three-dimensional time-dependent polychromatic transfer, PINNs produced low errors with small training times.
Results: The generalization-error bounds depend on computable training errors and quadrature-point counts, independently of the underlying dimension, and their predictions matched experiments.
Limitations: The inverse-problem treatment was brief, and careful analysis and practical applications of broader inverse extensions are deferred to future work.

A Estimates on the generalization error for the radiative transfer equation (1.1)

The appendix derives a rigorous generalization-error estimate for the trained PINN under assumptions on the radiative-transfer coefficients, scattering kernel, data, and quadrature rules. The proof connects the error to training residuals and quadrature approximations.

Assumptions: The analysis assumes a symmetric scattering kernel and boundedness of the associated function Ψ in L∞(S × Λ).
Assumptions: Lemma A.1 assumes bounded absorption and scattering coefficients, a sufficiently smooth symmetric kernel, a unique weak solution, and a smooth PINN output.
Assumptions: The generalization-error estimate requires Sobol training points, bounded Hardy-Krause variation of relevant data, and a Gauss quadrature rule of order s = s(ℓ).
Proof strategy: The proof defines the approximation error between the trained network and exact solution, derives an error equation, and estimates it through integration by parts, inequalities, and Grönwall’s inequality.
Conclusion: The final bound is obtained by inserting estimates for training and quadrature errors into the intermediate error relation.

B Estimates on the generalization error in the steady case

For the steady-state radiative transfer equation, the PINN is adapted by removing temporal dependence, and its generalization error is bounded under explicit regularity and coefficient assumptions. The bound depends on training errors and quadrature or sampling terms, with dimensionality-independent behavior when the PINN is trained well.

The steady-state equation is obtained by letting the speed of light c tend to infinity, removing temporal dependence from the PINN residuals and loss functions.
The analysis assumes a unique weak solution, bounded positive absorption and scattering coefficients, a symmetric smooth scattering kernel, and a bounded auxiliary function.
The steady-state generalization error is bounded in terms of interior and boundary training errors under Sobol sampling and a Gaussian quadrature rule for the scattering integral.
The proof derives an error equation for the difference between the PINN output and the exact solution, then estimates it using integration by parts, coefficient bounds, and boundary conditions.
Under the stated assumptions, the resulting bound implies that a well-trained PINN generalizes well and does not suffer from a curse of dimensionality.
The bound requires that the scattering coefficient not vary over too large a range unless sufficient absorption is present, while imposing no analogous scale restriction on absorption.