Abstract
Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations. With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. The choice of the loss function, which compares the training data trajectory in state space and the current estimated solution trajectory of the UDE to optimize the solution, plays a crucial role within this approach. The Mean Square Error as loss function contains the risk of a reconstruction which completely misses the dynamical behavior of the training data. By contrast, our suggested trajectory-based loss function which optimizes two largely independent components, the length and angle of state space vectors of the training data, performs reliable well in examples of systems from neuroscience, chemistry and biology showing Saddle-Node, Pitchfork, Hopf and Period-doubling bifurcations.
Highlights
Missing terms in dynamical systems are a challenging problem for modeling
In neural network regression models, which we use here to find an approximation to missing terms in the Universal Differential Equation (UDE), this is typically done by minimization of the Mean Squared Error (MSE) loss function[26]
Often, trained networks using MSE as loss function lead to a solution which is only slightly worse than the networks using the Length Difference and Angle Difference (LDA) loss function, but there is a risk of complete failure of the approximation depending on the network initialization
Summary
Missing terms in dynamical systems are a challenging problem for modeling. Recent developments in the combination of machine learning and dynamical system theory open possibilities for a solution. We show how physics-informed differential equations and machine learning—combined in the Universal Differential Equation (UDE) framework by Rackauckas et al.—can be modified to discover missing terms in systems that undergo sudden fundamental changes in their dynamical behavior called bifurcations With this we enable the application of the UDE approach to a wider class of problems which are common in many real world applications. Universal Differential Equations (UDE)[11] are a recently proposed method to learn dynamical systems from data with machine learning and can be combined with the Sparse Identification of Nonlinear Dynamics (SInDy) a lgorithm[12] to estimate an algebraic form of the dynamical system from data (see the section for details) We show how this approach can be applied to learn missing terms in systems that can undergo sudden fundamental changes in their dynamics called bifurcation. We set up a statistical comparison of how well the two loss functions perform in each of these systems in two different parameter regimes representing two different
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