We develop a systematic Bayesian framework for physics constrained parameter inference of stochastic differential equations (SDE) from partial observations. The physical constraints are derived for stochastic climate models but are applicable for many fluid systems. We first derive a condition for the Lyapunov stability of stochastic climate models based on energy conserva- tion. Stochastic climate models are globally stable when a quadratic form, which is related to the cubic nonlinear operator, is negative definite. We develop a new algorithm for the efficient sampling of negative definite matrices and also for imputing unobserved data which improve the accuracy of the parameter estimates. Our algorithm can efficiently be implemented on GPU architectures leading to significant speed ups. We evaluate the performance of our framework on two conceptual climate models.
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