ClimODE is a physics-informed neural ordinary differential equation (ODE) model for climate and weather forecasting. It addresses the limitations of traditional numerical simulations and data-driven black-box models by incorporating physical principles, such as advection, into its continuous-time dynamics. ClimODE learns global weather transport as a neural flow, enabling precise weather evolution and uncertainty quantification. The model uses a second-order neural continuity equation with inductive biases that ensure value-conserving dynamics, allowing for stable long-term forecasts. ClimODE outperforms existing data-driven methods in global and regional forecasting with a significantly smaller parameterization, achieving state-of-the-art performance. The model integrates local convolutions, long-range attention, and a Gaussian emission network to predict uncertainties and source variations. It is trained on a single GPU and provides an open-source PyTorch implementation. ClimODE demonstrates superior performance in global, regional, and climate forecasting tasks, outperforming competitive neural methods across various metrics and variables. The model incorporates a physics-based framework for weather modeling, enabling accurate predictions while accounting for uncertainties. ClimODE's performance is validated through extensive experiments, showing its effectiveness in capturing meteorological variables and improving climate forecasting. The model's ability to handle both local and global effects, along with its uncertainty estimation, makes it a promising approach for future climate and weather prediction.ClimODE is a physics-informed neural ordinary differential equation (ODE) model for climate and weather forecasting. It addresses the limitations of traditional numerical simulations and data-driven black-box models by incorporating physical principles, such as advection, into its continuous-time dynamics. ClimODE learns global weather transport as a neural flow, enabling precise weather evolution and uncertainty quantification. The model uses a second-order neural continuity equation with inductive biases that ensure value-conserving dynamics, allowing for stable long-term forecasts. ClimODE outperforms existing data-driven methods in global and regional forecasting with a significantly smaller parameterization, achieving state-of-the-art performance. The model integrates local convolutions, long-range attention, and a Gaussian emission network to predict uncertainties and source variations. It is trained on a single GPU and provides an open-source PyTorch implementation. ClimODE demonstrates superior performance in global, regional, and climate forecasting tasks, outperforming competitive neural methods across various metrics and variables. The model incorporates a physics-based framework for weather modeling, enabling accurate predictions while accounting for uncertainties. ClimODE's performance is validated through extensive experiments, showing its effectiveness in capturing meteorological variables and improving climate forecasting. The model's ability to handle both local and global effects, along with its uncertainty estimation, makes it a promising approach for future climate and weather prediction.