Discovering governing physical laws from noisy data is a grand challenge in many science and engineering research areas. We present a new approach to data-driven discovery of ordinary differential equations (ODEs) and partial differential equations (PDEs), in explicit or implicit form. We demonstrate our approach on a wide range of problems, including shallow water equations and Navier–Stokes equations. The key idea is to select candidate terms for the underlying equations using dimensional analysis, and to approximate the weights of the terms with error bars using our threshold sparse Bayesian regression. This new algorithm employs Bayesian inference to tune the hyperparameters automatically. Our approach is effective, robust and able to quantify uncertainties by providing an error bar for each discovered candidate equation. The effectiveness of our algorithm is demonstrated through a collection of classical ODEs and PDEs. Numerical experiments demonstrate the robustness of our algorithm with respect to noisy data and its ability to discover various candidate equations with error bars that represent the quantified uncertainties. Detailed comparisons with the sequential threshold least-squares algorithm and the lasso algorithm are studied from noisy time-series measurements and indicate that the proposed method provides more robust and accurate results. In addition, the data-driven prediction of dynamics with error bars using discovered governing physical laws is more accurate and robust than classical polynomial regressions.

Compressive sensing is used to determine the flow characteristics around a cylinder (Reynolds number and pressure/flow field) from a sparse number of pressure measurements on the cylinder. Using a supervised machine learning strategy, library elements encoding the dimensionally reduced dynamics are computed for various Reynolds numbers. Convex L₁ optimization is then used with a limited number of pressure measurements on the cylinder to reconstruct, or decode, the full pressure field and the resulting flow field around the cylinder. Aside from the highly turbulent regime (large Reynolds number) where only the Reynolds number can be identified, accurate reconstruction of the pressure field and Reynolds number is achieved. The proposed data-driven strategy thus achieves encoding of the fluid dynamics using the L₂ norm, and robust decoding (flow field reconstruction) using the sparsity promoting L₁ norm.

In this study, we applied an uncertainty quantification (UQ) technique to improve convective precipitation in the global climate model, the Community Atmosphere Model version 5 (CAM5), in which the convective and stratiform precipitation partitioning is very different from observational estimates. We examined the sensitivity of precipitation and circulation to several key parameters in the Zhang‐McFarlane deep convection scheme in CAM5, using a stochastic importance‐sampling algorithm that can progressively converge to optimal parameter values. The impact of improved deep convection on the global circulation and climate was subsequently evaluated. Our results show that the simulated convective precipitation is most sensitive to the parameters of the convective available potential energy consumption time scale, parcel fractional mass entrainment rate, and maximum downdraft mass flux fraction. Using the optimal parameters constrained by the observed Tropical Rainfall Measuring Mission, convective precipitation improves the simulation of convective to stratiform precipitation ratio and rain‐rate spectrum remarkably. When convection is suppressed, precipitation tends to be more confined to the regions with strong atmospheric convergence. As the optimal parameters are used, positive impacts on some aspects of the atmospheric circulation and climate, including reduction of the double Intertropical Convergence Zone, improved East Asian monsoon precipitation, and improved annual cycles of the cross‐equatorial jets, are found as a result of the vertical and horizontal redistribution of latent heat release from the revised parameterization. Positive impacts of the optimal parameters derived from the 2° simulations are found to transfer to the 1° simulations to some extent.