Structurally Aware Post-Training Quantization of Fourier Neural Operators

Fourier neural operators (FNOs) provide an efficient framework for learning mappings between function spaces, but deploying trained operators on resource-constrained hardware requires effective model compression. Post-training quantization (PTQ) is particularly relevant here because, unlike quantization-aware training (QAT), it requires no retraining and can be applied directly to a trained model. Despite this, the effectiveness of PTQ for neural operators remains largely unexplored. This paper presents a systematic study of PTQ for FNOs on two three-dimensional problems: forced isotropic turbulence and Darcy flow. We evaluate quantization strategies across model capacities and bit-widths at equal storage and find that quantizer design has a substantial impact on accuracy. In particular, different Fourier modes have very different sensitivity to quantization, and the measured input energy closely tracks quantization sensitivity. Allocating bits according to this input energy reduces error across all configurations tested, with a median improvement over uniform allocation of $5.2\times$ overall and of $14.9\times$, $5.9\times$ and $3.1\times$ at two, three and four bits. We also evaluate three other aspects of quantization: weight representation, scaling granularity, and quantization range. Once the bit budget is allocated according to input energy, these choices have substantially smaller effects. Nevertheless, at a fixed bit-width, different quantizer designs can produce errors that differ by up to two orders of magnitude. These results show that exploiting the structure of Fourier Neural Operators can substantially improve post-training quantization.