Continuous-time neural networks are a class of machine learning systems that can tackle representation learning on spatiotemporal decision-making tasks. These models are typically represented by continuous differential equations. However, their expressive power when they are deployed on computers is bottlenecked by numerical differential equation solvers. This limitation has notably slowed down the scaling and understanding of numerous natural physical phenomena such as the dynamics of nervous systems. Ideally, we would circumvent this bottleneck by solving the given dynamical system in closed form. This is known to be intractable in general. Here, we show that it is possible to closely approximate the interaction between neurons and synapses—the building blocks of natural and artificial neural networks—constructed by liquid time-constant networks efficiently in closed form. To this end, we compute a tightly bounded approximation of the solution of an integral appearing in liquid time-constant dynamics that has had no known closed-form solution so far. This closed-form solution impacts the design of continuous-time and continuous-depth neural models. For instance, since time appears explicitly in closed form, the formulation relaxes the need for complex numerical solvers. Consequently, we obtain models that are between one and five orders of magnitude faster in training and inference compared with differential equation-based counterparts. More importantly, in contrast to ordinary differential equation-based continuous networks, closed-form networks can scale remarkably well compared with other deep learning instances. Lastly, as these models are derived from liquid networks, they show good performance in time-series modelling compared with advanced recurrent neural network models.
States are using climate change adaptation plans to prepare for the impacts of climate change. As of October 2014, 14 states had finalized state-led adaptation plans. An additional eight states and the District of Columbia had some adaptation planning underway. The Georgetown Climate Center is tracking implementation of these plans and making these data available to scholars, practitioners, and the public through an interactive tool in the Climate Center's Adaptation Clearinghouse. This research provides, for the first time, an empirical assessment of the progress that states are making implementing their adaptation plans. Among states with plans, there is significant variation in the number of goals included in each plan, the sectors covered, and the degree of implementation that has occurred. The number of discrete goals in state plans ranges from 28 to 373, and the level of implementation progress ranges from 16 to 87%. Our research demonstrates that states are actively implementing the goals in their adaptation plans, but the process of plan development and the progress in implementation are highly state specific.
Continuous-depth neural models, where the derivative of the model's hidden state is defined by a neural network, have enabled strong sequential data processing capabilities. However, these models rely on advanced numerical differential equation (DE) solvers resulting in a significant overhead both in terms of computational cost and model complexity. In this paper, we present a new family of models, termed Closed-form Continuous-depth (CfC) networks, that are simple to describe and at least one order of magnitude faster while exhibiting equally strong modeling abilities compared to their ODE-based counterparts. The models are hereby derived from the analytical closed-form solution of an expressive subset of time-continuous models, thus alleviating the need for complex DE solvers all together. In our experimental evaluations, we demonstrate that CfC networks outperform advanced, recurrent models over a diverse set of time-series prediction tasks, including those with long-term dependencies and irregularly sampled data. We believe our findings open new opportunities to train and deploy rich, continuous neural models in resource-constrained settings, which demand both performance and efficiency.
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