Neural networks can be FLOP-efficient integrators of 1D oscillatory integrands

TL;DR

This work tackles efficient numerical integration of highly oscillatory 1D functions by training a feed-forward neural network to approximate $I=\int_{s_1}^{s_2} f(x)\,dx$ using fixed quadrature-point evaluations $f(x_i)$. The model outputs $\widehat{I}$ and is optimized under a FLOP-budget, achieving substantial FLOP-efficiency gains over traditional quadrature methods for sufficiently oscillatory inputs. Key findings indicate that networks with about 5 hidden layers balance accuracy (targeting NMSE around $10^{-3}$) and computational cost, with average FLOP gains near $10^3$ under suitable conditions. The approach leverages learned latent patterns in oscillatory integrands and is particularly advantageous in many-query, parametric settings, though it may not extrapolate well to integrands far outside the training distribution.

Abstract

We demonstrate that neural networks can be FLOP-efficient integrators of one-dimensional oscillatory integrands. We train a feed-forward neural network to compute integrals of highly oscillatory 1D functions. The training set is a parametric combination of functions with varying characters and oscillatory behavior degrees. Numerical examples show that these networks are FLOP-efficient for sufficiently oscillatory integrands with an average FLOP gain of 1000 FLOPs. The network calculates oscillatory integrals better than traditional quadrature methods under the same computational budget or number of floating point operations. We find that feed-forward networks of 5 hidden layers are satisfactory for a relative accuracy of 0.001. The computational burden of inference of the neural network is relatively small, even compared to inner-product pattern quadrature rules. We postulate that our result follows from learning latent patterns in the oscillatory integrands that are otherwise opaque to traditional numerical integrators.

Figures (7)

• Figure 1: (a) Newton--Cotes-like method of approximating an integral with a (b) feed-forward multi-layer perceptron neural network (NN). The model uses inputs $f(x_i)$, where the pseudo-quadrature points $x_i$ are fixed in the spatial domain, to compute the weights and biases of the neural networks, thus approximating $\widehat{I}$.
• Figure 2: Neural network (NN) model's performance on test and training dataset.
• Figure 3: Example canonical oscillatory test functions as labeled.
• Figure 4: Comparison of results of neural network (NN) model with Newton--Cotes methods for the oscillatory functions of \ref{['table2']}. The integrands associated with panels (a) and (b) are as shown in \ref{['fig4functions']}.
• Figure 5: Example test functions tested as labeled.