Feed-forward neural networks (FNNs) work as standard building blocks in applying artificial intelligence (AI) to the physical world. They allow learning the dynamics of unknown physical systems (e.g., biological and chemical) {to predict their future behavior}. However, they are likely to violate the physical constraints of those systems without proper treatment. This work focuses on imposing two important physical constraints: monotonicity (i.e., a partial order of system states is preserved over time) and stability (i.e., the system states converge over time) when using FNNs to learn physical dynamics. For monotonicity constraints, we propose to use nonnegative neural networks and batch normalization. For both monotonicity and stability constraints, we propose to learn the system dynamics and corresponding Lyapunov function simultaneously. As demonstrated by case studies, our methods can preserve the stability and monotonicity of FNNs and significantly reduce their prediction errors.
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