Finding parameters (values of coefficients) for a system of differential equations with constant coefficients at known values at a number of points.
- Finding parameters (values of coefficients) for a system of differential equations with constant coefficients at known values at a number of points.
- Consider the case where the observations of only one function from the system are known.
- Describe the dependence of the error on the maximum of the derivative on the segment for the special point of the center.
- Python 3 - used as main tool
- Scipy - for integrating a system of ordinary differential equations
- Matplotlib - for plotting graphs
- Numpy - arrays and various functionality
- Tensorflow (Keras submodule)
- Model - a model grouping layers into an object with training/inference features
- Layers
- Dense - just regular densely-connected NN layer
- Input - used to instantiate a Keras tensor
- Concatenate - layer that concatenates a list of inputs
- Optimizers
- Adam - Adam optimization is a stochastic gradient descent method that is based on adaptive estimation of first-order and second-order moments
Based on the findings, the following observations can be made regarding the dependence of the error on the maximum of the derivative on the segment for the special point of the center:
- In experiment 1, where the maximum of the derivative is smaller, the error (MSE) is 0.0158 at the predicted parameters (with noise) [-2.0177553, 8.126868] and 0.0158 at the predicted parameters (without noise) [-1.8328079, 7.384569].
- In experiment 2, where the maximum of the derivative is also small, the error is 0.0057 at the predicted parameters (with noise) [-2.0182137, 8.076085] and 0.0057 at the predicted parameters (without noise) [-2.1121407, 8.456265].
- In experiment 3, where the maximum of the derivative increases, the error increases and is 0.0445 at the predicted parameters (with noise) [-1.9907327, 8.210772] and 0.0445 at the predicted parameters (without noise) [-1.9635643, 8.092097].
- Other experiments also show an increase in the error as the maximum of the derivative on the segment for the special point of the center increases. This indicates that the model is less accurate in predicting the parameters of the equation when the derivative of the function on the segment becomes more complex or rapidly increasing.
In general, we can conclude that the dependence of the error on the maximum of the derivative on the segment for the special point of the center indicates that the model may have difficulty in accurately predicting the parameters of the equation when the derivatives of the function change rapidly or have high values. This may require more complex models or more data for more accurate predictions in such cases.
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