  Constructing accurate and generalizable approximators for complex
physico-chemical processes exhibiting highly non-smooth dynamics is
challenging. In this work, we propose new developments and perform comparisons
for two promising approaches: manifold-based polynomial chaos expansion (m-PCE)
and the deep neural operator (DeepONet), and we examine the effect of
over-parameterization on generalization. We demonstrate the performance of
these methods in terms of generalization accuracy by solving the 2D
time-dependent Brusselator reaction-diffusion system with uncertainty sources,
modeling an autocatalytic chemical reaction between two species. We first
propose an extension of the m-PCE by constructing a mapping between latent
spaces formed by two separate embeddings of input functions and output QoIs. To
enhance the accuracy of the DeepONet, we introduce weight self-adaptivity in
the loss function. We demonstrate that the performance of m-PCE and DeepONet is
comparable for cases of relatively smooth input-output mappings. However, when
highly non-smooth dynamics is considered, DeepONet shows higher accuracy. We
also find that for m-PCE, modest over-parameterization leads to better
generalization, both within and outside of distribution, whereas aggressive
over-parameterization leads to over-fitting. In contrast, an even highly
over-parameterized DeepONet leads to better generalization for both smooth and
non-smooth dynamics. Furthermore, we compare the performance of the above
models with another operator learning model, the Fourier Neural Operator, and
show that its over-parameterization also leads to better generalization. Our
studies show that m-PCE can provide very good accuracy at very low training
cost, whereas a highly over-parameterized DeepONet can provide better accuracy
and robustness to noise but at higher training cost. In both methods, the
inference cost is negligible.
