Bitwise efficiency in chaotic models

Motivated by the increasing energy consumption of supercomputing for weather and climate simulations, we introduce a framework for investigating the bit-level information efficiency of chaotic models. In comparison with previous explorations of inexactness in climate modelling, the proposed and tested information metric has three specific advantages: (i) it requires only a single high-precision time series; (ii) information does not grow indefinitely for decreasing time step; and (iii) information is more sensitive to the dynamics and uncertainties of the model rather than to the implementation details. We demonstrate the notion of bit-level information efficiency in two of Edward Lorenz's prototypical chaotic models: Lorenz 1963 (L63) and Lorenz 1996 (L96). Although L63 is typically integrated in 64-bit 'double' floating point precision, we show that only 16 bits have significant information content, given an initial condition uncertainty of approximately 1% of the size of the attractor. This result is sensitive to the size of the uncertainty but not to the time step of the model. We then apply the metric to the L96 model and find that a 16-bit scaled integer model would suffice given the uncertainty of the unresolved sub-grid-scale dynamics. We then show that, by dedicating computational resources to spatial resolution rather than numeric precision in a field programmable gate array (FPGA), we see up to 28.6% improvement in forecast accuracy, an approximately fivefold reduction in the number of logical computing elements required and an approximately 10-fold reduction in energy consumed by the FPGA, for the L96 model.