Dynamical systems describe how a physical system evolves over time. Physical processes can evolve faster or slower in different environmental conditions. We use time-warping as rescaling the time in a model of a physical system. This thesis proposes a new method of transfer learning for Recurrent Neural Networks (RNNs) based on time-warping. We prove that for a class of linear, first-order differential equations known as time lag models, an LSTM can approximate these systems with any desired accuracy, and the model can be time-warped while maintaining the approximation accuracy. The Time-Warping method of transfer learning is then evaluated in an applied problem on predicting fuel moisture content (FMC), an important concept in wildfire modeling. An RNN with LSTM recurrent layers is pretrained on fuels with a characteristic time scale of 10 hours, where there are large quantities of data available for training. The RNN is then modified with transfer learning to generate predictions for fuels with characteristic time scales of 1 hour, 100 hours, and 1000 hours. The Time-Warping method is evaluated against several known methods of transfer learning. The Time-Warping method produces predictions with an accuracy level comparable to the established methods, despite modifying only a small fraction of the parameters that the other methods modify.

Data-driven artificial intelligence (AI) models have made significant advancements in weather forecasting, particularly in medium-range and nowcasting. However, most data-driven weather forecasting models are black-box systems that focus on learning data mapping rather than fine-grained physical evolution in the time dimension. Consequently, the limitations in the temporal scale of datasets prevent these models from forecasting at finer time scales. This paper proposes a physics-AI hybrid model (i.e., WeatherGFT) which generalizes weather forecasts to finer-grained temporal scales beyond training dataset. Specifically, we employ a carefully designed PDE kernel to simulate physical evolution on a small time scale (e.g., 300 seconds) and use a parallel neural networks with a learnable router for bias correction. Furthermore, we introduce a lead time-aware training framework to promote the generalization of the model at different lead times. The weight analysis of physics-AI modules indicates that physics conducts major evolution while AI performs corrections adaptively. Extensive experiments show that WeatherGFT trained on an hourly dataset, effectively generalizes forecasts across multiple time scales, including 30-minute, which is even smaller than the dataset's temporal resolution.

The dense connections between layers change the definition of what from one layer to the next.

Time perception is fundamental in our daily life. An important feature of time perception is temporal scaling (TS): the ability to generate temporal sequences (e.g., movements) with different speeds. However, it is largely unknown about the mathematical principle underlying TS in the brain.

Inference in the introduced linear Gaussian SSM is straightforward and can be performed using Kalman prediction and observation updates.

We show it depends on the precise way in which the limit is taken, and in particular on how the quantityofdata,thehiddenlayerwidth,&thelearningratescalesasd .

The dynamics of gradient-based training in neural networks often exhibit nontrivial structures; hence, understanding them remains a central challenge in theoretical machine learning. In particular, a concept of feature unlearning, in which a neural network progressively loses previously learned features over long training, has gained attention. In this study, we consider the infinite-width limit of a two-layer neural network updated with a large-batch stochastic gradient, then derive differential equations with different time scales, revealing the mechanism and conditions for feature unlearning to occur. Specifically, we utilize the fast-slow dynamics: while an alignment of first-layer weights develops rapidly, the second-layer weights develop slowly. The direction of a flow on a critical manifold, determined by the slow dynamics, decides whether feature unlearning occurs. We give numerical validation of the result, and derive theoretical grounding and scaling laws of the feature unlearning. Our results yield the following insights: (i) the strength of the primary nonlinear term in data induces the feature unlearning, and (ii) an initial scale of the second-layer weights mitigates the feature unlearning.