Towards Stability of Autoregressive Neural Operators

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and memory expense -- these systems are often forced to rely on autoregressive time-stepping of the neural network to predict future temporal states. While this is effective in managing costs, it can lead to uncontrolled error growth over time and eventual instability. We analyze the sources of this autoregressive error growth using prototypical neural operator models for physical systems and explore ways to mitigate it. We introduce architectural and application-specific improvements that allow for careful control of instability-inducing operations within these models without inflating the compute/memory expense. We present results on several scientific systems that include Navier-Stokes fluid flow, rotating shallow water, and a high-resolution global weather forecasting system. We demonstrate that applying our design principles to prototypical neural networks leads to significantly lower errors in long-range forecasts with 800\% longer forecasts without qualitative signs of divergence compared to the original models for these systems. We open-source our \href{https://anonymous.4open.science/r/stabilizing_neural_operators-5774/}{code} for reproducibility.

Learning to Assimilate in Chaotic Dynamical Systems

The accuracy of simulation-based forecasting in chaotic systems is heavily dependent on high-quality estimates of the system state at the time the forecast is initialized. Data assimilation methods are used to infer these initial conditions by systematically combining noisy, incomplete observations and numerical models of system dynamics to produce effective estimation schemes. We introduce amortized assimilation, a framework for learning to assimilate in dynamical systems from sequences of noisy observations with no need for ground truth data. We motivate the framework by extending powerful results from self-supervised denoising to the dynamical systems setting through the use of differentiable simulation. Experimental results across several benchmark systems highlight the improved effectiveness of our approach over widely-used data assimilation methods.

Multigrid for Bundle Adjustment

Bundle adjustment is an important global optimization step in many structure from motion pipelines. Performance is dependent on the speed of the linear solver used to compute steps towards the optimum. For large problems, the current state of the art scales superlinearly with the number of cameras in the problem. We investigate the conditioning of global bundle adjustment problems as the number of images increases in different regimes and fundamental consequences in terms of superlinear scaling of the current state of the art methods. We present an unsmoothed aggregation multigrid preconditioner that accurately represents the global modes that underlie poor scaling of existing methods and demonstrate solves of up to 13 times faster than the state of the art on large, challenging problem sets.