Beyond Affine Reconstruction: Multi-Resolution Transformer DeepONets for the Radiative Transfer Equation

Abstract

Radiative heat transfer plays a dominant role in fire simulations, where solving the radiative transfer equation (RTE) is a computational bottleneck. Neural operators can accelerate these computations by learning solution operators from offline simulation data. However, conventional DeepONet surrogates rely on fixed-grid inputs and affine reconstruction, limiting their flexibility across resolutions and their approximation accuracy at a fixed latent dimension. In this paper, we develop a transformer-based DeepONet for multi-resolution RTE surrogate modelling. The proposed approach combines a Perceiver encoder, which maps temperature and absorption coefficient fields at different spatial resolutions into a fixed-size latent representation, with a cross-attention decoder that enables non-affine reconstruction of the radiative intensity. This architecture allows a single model to process inputs at varying resolutions without resampling them onto a common grid. We formulate a computable empirical version of a previously derived theoretical lower bound for affine reconstruction, enabling direct comparison with model errors on the same dataset. Experiments on a two-dimensional pool-fire dataset demonstrate that the proposed approach outperforms autoencoder-based DeepONets in predicting both radiative intensity and incident radiation, with fewer trainable parameters. Power spectral density analysis shows improved preservation of small-scale structures. Both affine baselines remain above the corresponding lower bound at all evaluated ranks, whereas the proposed model achieves a training error below it at one matched latent dimension, demonstrating the potential of non-affine reconstruction to overcome this representational limitation. Experiments on a dataset combining three spatial resolutions confirm the accuracy advantage of the approach at each input resolution.