Table 4 Linear damped oscillator: the discovered governing equations using IRK-SINDy in the approach of Newton’s iterations and fixed point iterations for various noise levels \(\sigma\).
Noise | Newton’s iterations | Fixed point iterations |
|---|---|---|
\(\sigma = 0.01\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.102 x_{1}(t) + 2.000 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.099 x_{2}(t) \end{array}\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.102 x_{1}(t) + 2.000 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.099 x_{2}(t) \end{array}\) |
\(\sigma = 0.04\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.105 x_{1}(t) + 1.995 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.009 x_{1}(t) - 0.097 x_{2}(t) \end{array}\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.105 x_{1}(t) + 1.995 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.008 x_{1}(t) - 0.097 x_{2}(t) \end{array}\) |
\(\sigma = 0.08\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.117 x_{1}(t) + 2.013 x_{2}(t)\\ & \dot{x}_{2}(t) = -1.972 x_{1}(t) - 0.105 x_{2}(t) \end{array}\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.117 x_{1}(t) + 2.013 x_{2}(t)\\ & \dot{x}_{2}(t) = -1.972 x_{1}(t) - 0.105 x_{2}(t) \end{array}\) |
\(\sigma = 0.16\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.147 x_{1}(t) + 1.991 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.006 x_{1}(t) - 0.086 x_{2}(t) \end{array}\) | \(\begin{array}{cc} & \dot{x}_{1}(t) = -0.148 x_{1}(t) + 1.991 x_{2}(t)\\ & \dot{x}_{2}(t) = -2.006 x_{1}(t) - 0.088 x_{2}(t) \end{array}\) |
