Modelling in Engineering - Lecture 1

These flashcards cover model classification, fidelity levels, complexity, and systems identification principles from the first lecture of the Test and Simulation for Gas Turbines course.

Deterministic model

A model where the outcome is certain given initial values and boundary conditions; a given input always leads to the same output.

Stochastic model

A model that includes a random or probabilistic component where outputs are random variables that can take on values in an uncertainty range; the same input can lead to different outputs.

Statistical data-driven modelling

A non-deterministic approach that combines mathematical modelling and statistical assumptions to represent a set of data, assuming the data comes from a greater population through sampling.

Training data set

Also called estimation data in system identification, it is the data set used to learn the statistical model.

Test data set

Also called validation data in system identification, it is the data set used to test the model's ability to make correct predictions on a new set of data.

Accuracy

How close the model results are to the actual value of the real system.

Precision

How close the model results are to each other.

Model fidelity

A measure of how accurately a model describes a real system in terms of the amount of features captured and the accuracy of results compared to real values.

L0 Fidelity

The lowest level of fidelity, based on empirical knowledge or rough estimations.

L1 Fidelity

A fidelity level using simplified physics principles with a limited number of effects considered.

L2 Fidelity

A fidelity level where physical behaviour is represented by more accurate relationships and equations.

L3 Fidelity

The highest fidelity level, involving very high computational costs to capture local, distinct effects and a high number of features with high precision.

Computational complexity

Also known as algorithm-sided complexity, it refers to hardware-related requirements such as computational time and memory storage, often expressed via Big-O-Notation.

Big-O-Notation

A measure of computational complexity where $O$ is the order of the function and $n$ is the number of elements, e.g., $O(n^2)$ or $O(\text{log}(n)).$

Problem-sided complexity

Model-related complexity concerning the model structure, including the number of parameters, degree of polynomial, or features used.

Underfitting

A condition where a model is 'too simple' and cannot capture relevant effects because important physical effects were neglected or the polynomial degree is too low.

Overfitting

A condition where a model is 'too complex' and captures irrelevant information like noise, meaning the model is over-trained and cannot generalise to a test dataset.

Cross-validation error

An indicator for generalisation quality calculated as $\varepsilon_{cv} = \frac{1}{n}\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$ by randomly splitting training data into subsets.

Degree of abstraction

Refers to how far a physics-based model distances itself from reality in terms of physical insight.

White-box modelling

Modelling based on a complete understanding of the inner system structure and physical law-driven equations.

Gray-box modelling

Modelling used when knowledge of the inner system structure is limited; it captures essential system information and is partly data-driven.

Black-box modelling

Fully data-driven modelling used when no knowledge of the inner system structure is available, focusing only on input-output (IO) behaviour.

Parametric models

Models based on a finite set of parameters where the response is a function of input variables and parameters, such as a polynomial structure like $y = p_1x + p_0.$

Non-parametric models

Models where the parameter space is infinite-dimensional and no assumptions are made on data distribution, such as k-nearest neighbours (kNN).

Continuous models

Models where output changes continuously over time, often represented by sets of ordinary differential equations (ODEs).

Discrete models

Computer-based models where the output changes only at discrete points in time, such as a state machine.

System identification

The process of deriving a model of a dynamic system from data by minimising the error between the model output $\hat{y}$ and the real system output $y.$