Table 1 Linear damped oscillator: the discovered governing equations using deep IRK-SINDy, RK4-SINDy, and Conv-SINDy for various sample-size m.

From: Implicit Runge-Kutta based sparse identification of governing equations in biologically motivated systems

Sample-size

Deep IRK-SINDy

RK4-SINDy

Conv-SINDy

\(m = 801\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.100 x_{1}(t) + 2.000 x_{2}(t)\\ \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.100 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.100 x_{1}(t) + 2.000 x_{2}(t)\\ \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.100 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.100 x_{1}(t) + 2.000 x_{2}(t)\\ \dot{x}_{2}(t) = -2.000 x_{1}(t) - 0.100 x_{2}(t) \end{array}\)

\(m = 201\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.100 x_{1}(t) + 2.000 x_{2}(t)\\ \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.100 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.100 x_{1}(t) + 2.001 x_{2}(t)\\ \dot{x}_{2}(t) = -2.001 x_{1}(t) - 0.100 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.099 x_{1}(t) + 1.987 x_{2}(t)\\ \dot{x}_{2}(t) = -1.989 x_{1}(t) - 0.098 x_{2}(t) \end{array}\)

\(m = 41\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.101 x_{1}(t) + 2.003 x_{2}(t)\\ \dot{x}_{2}(t) = -2.003 x_{1}(t) - 0.101 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.103 x_{1}(t) + 2.011 x_{2}(t)\\ \dot{x}_{2}(t) = -2.011 x_{1}(t) - 0.103 x_{2}(t) \end{array}\)

\(\begin{array}{cc} & \dot{x}_{1}(t) = -0.066 x_{1}^2(t) - 0.083 x_{1}^{3}(t)\\ & + 1.680 x_{2}(t)\\ & \dot{x}_{2}(t) = -1.545 x_{1}(t) - 0.080 x_{1}^{2}(t)\\ & - 0.140 x_{1}^{3}(t) + 0.063 x_{1}^{2}(t)x_{2}(t)\\ & - 0.096 x_{2}(t) \end{array}\)

\(m = 31\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.102 x_{1}(t) + 2.009 x_{2}(t)\\ \dot{x}_{2}(t) = -2.008 x_{1}(t) - 0.103 x_{2}(t) \end{array}\)

\(\begin{array}{cc} \dot{x}_{1}(t) = -0.107 x_{1}(t) + 2.017 x_{2}(t)\\ \dot{x}_{2}(t) = -2.016 x_{1}(t) - 0.106 x_{2}(t) \end{array}\)

\(\begin{array}{cc} & \dot{x}_{1}(t) = -0.158 x_{1}(t) -0.099 x_{1}^{2}(t)\\ & -0.225 x_{1}^{3}(t) -0.125 x_{1}(t)x_{2}(t)\\ & + 0.103 x_{1}(t)x_{2}^{2}(t) + 1.424 x_{2}(t)\\ & + 0.050 x_{2}^{3}(t)\\ & \dot{x}_{2}(t) = -1.309 x_{1}(t) -0.098 x_{1}^{2}(t)\\ & - 0.213 x_{1}^{3}(t) -0.126 x_{1}(t)x_{2}(t)\\ & + 0.121 x_{1}(t)x_{2}^{2}(t) - 0.084 x_{2}(t)\\ & + 0.062 x_{2}^{3}(t) \end{array}\)

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