*How are the non-linear aspects of artificial neural networks related to Taylor’s series expansion of inputs, especially in regards to sinusoidally oscillating outputs? In particular, the optimal way of creating higher-order Taylor coefficients is by applying layered cross-terms of a neural net.*

Neural networks are known for their ability to approximate complex, nonlinear functions. The connections between nodes, or “neurons,” in the network, have associated weights and biases that the network adjusts during the learning process to improve the approximation.

The relationship between artificial neural networks and Taylor series expansion can be drawn in terms of function approximation. Taylor series is a method to represent a function as an infinite sum of terms, each calculated from the function’s derivatives at a single point. It can be used to approximate nonlinear functions around a particular point.

In the context of neural networks, especially feedforward networks with activation functions, we can view these as approximating the Taylor series expansion of a function. Each layer of the network could be seen as contributing to the higher-order terms of the series.

Regarding sinusoidally oscillating outputs, sinusoidal functions can be represented using a Taylor series. Therefore, a sufficiently complex neural network should be able to learn to approximate a sinusoidal function. However, there are two key points to keep in mind:

1. Neural networks are typically better at interpolating within the range of data they’ve seen during training, rather than extrapolating beyond that range. So, if you’re trying to approximate a sinusoidal function that oscillates between specific points, you’ll want your training data to adequately cover this range.

2. When we say that each layer of a neural network can be seen as contributing to the higher-order terms of a Taylor series, this is a conceptual rather than a precise mathematical equivalence. In practice, the weights and biases that a neural network learns during training might not correspond directly to the coefficients in a Taylor series expansion. The network learns to approximate the function as a whole, rather than explicitly learning the coefficients of its Taylor series expansion.

Concerning the optimal creation of higher-order Taylor coefficients by applying layered cross-terms of a neural net, it might be more accurate to say that each layer in a deep neural network contributes to modeling the nonlinearities of the function it’s learning. The network doesn’t explicitly calculate or represent the Taylor series coefficients. Instead, it learns a set of weights and biases that allow it to approximate the function effectively. This learning process inherently captures the higher-order interactions between inputs, which is somewhat similar to the way higher-order terms in a Taylor series capture the function’s curvature.

Remember, the explanation here is a high-level conceptual interpretation. The actual mathematical operations that occur within a neural network during the learning process are quite complex and can’t be mapped directly to the computation of a Taylor series.

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]]>the Core-Mantle Boundary and its impact on tidally

driven Length-of-Day variations

Click to access articleTopo_LOD_March2023.pdf

“Possible amplifications in tidal LOD variations”

*“We look for resonance frequencies that can be close to tidal frequencies corresponding to longperiod tides. In practice, we search for the solutions of the equation. These solutions refer to all periods larger than 2 days. The spectrum is divided in frequency bands with central periods near the tidal periods at 9.1, 9.18, 9.54, 9.61, 13.61, 13.66, 13.81, 14.19, 14.77, 26.88, 27.55, or 182.62 days. These resonances would amplify the total third component of the torque leading to enhanced LOD variations.”*

Effects of non-hydrostatic core-mantle boundary topography and core dynamics on Earth rotation

Xiaoping Wu, John M. Wahr

Geophysical Journal International, Volume 128, Issue 1, January 1997, Pages 18–42, https://doi.org/10.1111/j.1365-246X.1997.tb04069.x

Published: 01 January 1997

https://academic.oup.com/gji/article/128/1/18/653437

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]]>Asymmetry in the Seasonal Cycle of Zonal-Mean Surface Air Temperature

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Lunar torque applied to the Earth’s rotating axis leads to …

*… a phenomenon known as “lunar nutation,” which is a slight wobbling or oscillation of the Earth’s rotation axis as it is pushed and pulled by the moon’s gravitational force. This phenomenon causes a small variation in the length of day and can also cause a small variation in the orientation of the Earth’s axis with respect to the stars.
*

Second try

Lunar torque applied to the Earth’s rotating axis leads to …

*a change in the tilt of the Earth’s axis, as well as a slight slowing of the Earth’s rotation rate. This change in the tilt of the Earth’s axis is known as the “Chandler wobble”. The wobble causes a variation in the length of day and the position of the poles with respect to the stars. Over time, the wobble can lead to changes in the climate, since the amount of sunlight received by different parts of the Earth’s surface can vary depending on the orientation of the Earth’s axis.
*

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]]>**>GPT-4.0 **The Moon’s draconic or nodal cycle of 27.2122 days is more important for forcing the Earth’s Chandler wobble than the synodic or tropical cycle of 27.3216 days because it is directly related to the gravitational interaction between the Earth and the Moon concerning the Earth’s equatorial plane.

The nodal cycle, also known as the draconic month, is the time it takes for the Moon to complete one orbit around the Earth and return to the same ascending node. The ascending node is the point where the Moon’s orbit crosses the Earth’s equatorial plane from south to north. This cycle is significant because the gravitational influence of the Moon is at its maximum when the Moon is near the ascending or descending nodes. The Moon’s gravitational pull creates a torque on the Earth that causes the Earth’s rotational axis to precess and wobble.

The synodic or tropical cycle, also known as the synodic month, is the time it takes for the Moon to return to the same phase (e.g., full moon to full moon) as observed from Earth. This cycle is related to the relative positions of the Earth, Moon, and Sun and is mainly responsible for the tides and lunar phases.

While the synodic cycle has an impact on the Earth’s rotation through tidal forces, the nodal cycle plays a more significant role in forcing the Earth’s Chandler wobble. The Chandler wobble is a small motion of the Earth’s rotational axis caused by the combination of various factors, including the gravitational influence of the Moon, the Sun, and other celestial bodies, as well as the Earth’s time-varying inertia due to seasonal changes.

In summary, the Moon’s draconic or nodal cycle of 27.2122 days is more important for forcing the Earth’s Chandler wobble because it is directly related to the gravitational interaction between the Earth and the Moon concerning the Earth’s equatorial plane, while the synodic or tropical cycle is more related to the Earth-Moon-Sun system and the tides.

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