Deep Learning for Stable Monotone Dynamical Systems

Monotone systems, originating from real-world (e.g., biological or chemical) applications, are a class of dynamical systems that preserves a partial order of system states over time. In this work, we introduce a feedforward neural networks (FNNs)-based method to learn the dynamics of unknown stable nonlinear monotone systems. We propose the use of nonnegative neural networks and batch normalization, which in general enables the FNNs to capture the monotonicity conditions without reducing the expressiveness. To concurrently ensure stability during training, we adopt an alternating learning method to simultaneously learn the system dynamics and corresponding Lyapunov function, while exploiting monotonicity of the system.~The combination of the monotonicity and stability constraints ensures that the learned dynamics preserves both properties, while significantly reducing learning errors. Finally, our techniques are evaluated on two complex biological and chemical systems.