Fig. 4 | Scientific Reports

Fig. 4

From: Physics-informed neural networks with hybrid Kolmogorov-Arnold network and augmented Lagrangian function for solving partial differential equations

Fig. 4

The frequency histogram statistics show the optimal Lagrangian multipliers \(\lambda ^{*}\) in all benchmark experiments. Where the Helmholtz equation contains \(\lambda ^{*}_{1}\), the Burgers equation contains \(\lambda ^{*}_{1},\lambda ^{*}_{2}\), the Black-Scholes equation contains \(\lambda ^{*}_{1},\lambda ^{*}_{2},\lambda ^{*}_{3}\), and the Klein-Gordon equation contains \(\lambda ^{*}_{1},\lambda ^{*}_{2},\lambda ^{*}_{3}\).

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