Multifidelity Kolmogorov-Arnold Networks

Abstract

We develop a method for multifidelity Kolmogorov-Arnold networks (KANs), which use a low-fidelity model along with a small amount of high-fidelity data to train a model for the high-fidelity data accurately. Multifidelity KANs (MFKANs) reduce the amount of expensive high-fidelity data needed to accurately train a KAN by exploiting the correlations between the low- and high-fidelity data to give accurate and robust predictions in the absence of a large high-fidelity dataset. In addition, we show that multifidelity KANs can be used to increase the accuracy of physics-informed KANs (PIKANs), without the use of training data.

Figures (13)

• Figure 1: Example multifidelity KAN. The low-fidelity data is given by $f_L(x) = x^2+ \sin(20x)$ and $f_H(x) = f_L(x)-x+\sin(5x)$ for $x\in[0, 1]$. It is evident that the linear KAN learns the linear correlations and the nonlinear KAN outputs the nonlinear correlation ($\sin(5x)$).
• Figure 2: Results for Test 1 with $N_{LF} = 50$. a) Plotted low- and high-fidelity data points and functions for reference. b) Loss values for multifidelity, low-fidelity, and high-fidelity training. c) Low-fidelity reference function and low-fidelity prediction. d) High-fidelity reference function and multifidelity and high-fidelity predictions.
• Figure 3: Results for Test 1 with $N_{LF} = 300$. a) Plotted low- and high-fidelity data points and functions for reference. b) Loss values for multifidelity, low-fidelity, and high-fidelity training. c) Low-fidelity reference function and low-fidelity prediction. d) High-fidelity reference function and multifidelity and high-fidelity predictions.
• Figure 4: Results for Test 2. a) Reference solutions and training data. b) Loss curves for single-fidelity and multifidelity training. c) Low-fidelity reference solution and prediction. d) High-fidelity reference solution and multifidelity prediction with $w=1.$ e) High-fidelity reference solution and multifidelity prediction with $w=0.$, f) High-fidelity reference solution and high-fidelity prediction.
• Figure 5: Results for Test 3. a) Depiction of the high-fidelity data locations. b) Loss curves for low-fidelity, high-fidelity, and multifidelity training. c) Trained predictions along the line $y = 0.67$. d) Trained predictions along the line $x = 0.67$.