Everybody uses modeling and simulation, even without being aware of doing it!

Transcription

1 Everybody uses modeling and simulation, even without being aware of doing it!

2 What is this course about? Modeling as a general tool to solve complex scientific / engineering problems, not just a mathematical equation Computational modeling & simulation to solve complex problems & to reduce experimental costs A better exploration of the information content of experimental data Answer questions such as What is a mathematical model? What types of models do exist? Which model is appropriate for a particular problem? How does one set up a mathematical model? What is simulation, parameter estimation, validation?

3 Course Content Mechanistic Models Differential equations Variational Principle Solution to ODEs Classical ODE systems Cellular automata Others: inverse problems, difference equation, etc Phenomenological Models Regression Neural networks Others: power law, networks, etc

4 Course Assignment & Evaluation I hope it to help your research, now or in the future Course project (term project throughout the course): 60% Select a problem in your area of interest Progress report (5%) Midterm report + presentation (20%) Final report + presentation (35%) Critical paper review (20%) Homework (10%) Participation (10%)

5 C1 Principles of Mathematical Modeling : Computational Modeling and Simulation Instructor: Linwei Wang

6 A Complex World Needs Models Engineers/scientists try to understand, develop, or optimize complex systems System: the object of interest. Plant cell, an atom, a organ, an electric circuit, a galaxy, To break the complexity of a system, simplified descriptions of the system (i.e., models) are needed to make the problem tractable Examples: a car that refuses to start

7 A Complex World Needs Models Problem: a car that refuses to start What is the system? What is the model (assuming we know to check the fuel & battery from experience)? Tank Ba.ery Car as a system Car as a model

8 What is a Model? To an observer B, an object A* is a model of an object A to the extent that B can use A* to answer questions that interest him about A Goal driven What is a good model? The best model is the simplest model that still serves its purpose, that is, which is still complex enough to help us understand a system and to solve problems. Seen in terms of a simple model, the complexity of a complex system will no longer obstruct our view, and we will virtually be able to look through the complexity of the system at the heart of things

9 Modeling and Simulation Scheme Definitions Of a problem to be solved or a question to be answered Of a system, i.e., a part of reality that pertains to the problem or question Systems Analysis (literature; experimental design) Identify parts of the system that are relevant for the problem or question Modeling Develop a model of the system based on the systems analysis Simulation Apply the model to the problem or question Derive a strategy to solve the problem or answer the question Validation (model-data fit does not guarantee this) Does the strategy derived solve the problem or answer the question?

10 What is Simulation & System? Simulation The application of a model with the objective to derive strategies that help solve a problem or answer a question pertaining to the system Simulare: to pretend (The model pretends to be the system) System Any kind of object to which we have a question that can be answered mathematically

11 Mathematics as a Natural Modeling Input-output systems Language Must be observable Transform input parameters to output parameters Can be naturally seen as a mathematical function y=f(x) A simplified representation of the real system in mathematical terms The mapping of the internal mechanics of real systems into mathematical operations Proved to be extremely fruitful to the understanding, optimization, or development of systems

12 What is a Mathematical Model? A mathematical model is an abstract, simplified, mathematical construct related to a part of reality and created for a particular purpose A triplet (S, Q, M) S: system Q: question relating to S M: a set of mathematical statement can be used to answer Q Importance of Q Example: mechanical system S Q: behavior of S at moderate velocity: M = {Newtonian mechanics} Q: behavior of S at velocity close to the speed of light: M = {relativistic mechanics}

13 A Very Simple Example We want to minimize the material consumption to build a cylindrical tin having a volume of 1 L S: the cylindrical tin Q: minimize the material consumption M: {!r 2 h =1, min(2!r 2 + 2!rh) } r,h

14 State Variables and System Parameters Reduced system S r The description of S in terms of the mathematical model The cylindrical tin example, Sr = {r, h} Neglecting other factors such as the thickness, material, color, roughness, etc, of the metal sheets State variables Describe the state of S in terms of M and are required to answer Q: describe the system properties we are interested in Tin example: s = A System parameters Mathematical quantities that describe the properties of S in terms of M, and are needed to compute the state variables: describe the system properties needed to obtain the state variables mathematically Tin example: p = {r, h}

15 Another Simple Example To predict the time evolution of the overall biomass of a plant System: plant Model: m = e r (r: growth rate) Reduced system: S r = {r} Justify your Sr The details of S r depend on the question Q to be answered. Ideally it will consist of no more than exactly those properties of the original system that are important to answer the Q being investigated Model guides the experiments: Typically parameters of Sr are those that need experimental characterization

16 Mathematical Models as Door Opener Translating originally non-mathematical problems into the language of mathematics, where powerful mathematical methods become applicable System S Ques5on Q Mathema5cal Model (S, Q, M) Answer A Real world Mathema5cal Problem M Answer A* Mathema.cs

17 How to Set Up Simple Models Define the unknowns Translate the problem into mathematical formulations Start with the simplest models!

18 What Types of Models Exist? Phenomenological models vs. Mechanistic models Phenomenal model Constructed based on experimental data only, using no a priori information about S Treat S as a black box AKA Empirical models, statistical models, data-driven models, blackbox model, etc Mechanistic model Some of the statements in M are based on a priori information about S Treat S as a white box or, often times, a grey box

19 Another Simple Example System 1 with input x (N) and output y (cm) x System 1 y X (N) y(cm) Black- box system Input- output data Phenomenological model Data fitting: y = ax + b a = 0.33, b = -0.5

20 The Same Simple Example Now if we assume that the internal mechanics of system a looks like this: y = kx, estimate k from the data

21 Phenomenological vs. Mechanistic Models Mechanistic models: Allow better predictions of system behavior Phenomenological models are expected to be valid only close to the range of the data Allow better predictions of modified system behaviors Consider replacing the previous system by one that has 2 springs Physically interpretable parameters Particularly important when we want to optimize system performance Phenomenological models: Universally applicable Mechanistic models require a priori knowledge of the system Might be more efficient when dealing with a very complex system Basically explains the popularity of neural network methods No risk of wrong prior knowledge

22 More Classification of Models Stationary vs. instationary models Whether at least one of the system parameters or state variables depends on time Distributed vs. lumped models Whether at least one of the system parameters or state variables depends on space variables Direct vs. inverse models

23 From Black to White Box Models Electrical circuits Mechanical Chemical Biological Economic Social Psychological Design Control Analysis Predic5on Specula5on White box Black box

24 Don ts of Mathematical Modeling Don t believe that the model is the reality Don t extrapolate beyond the region of fit Don t distort reality to fit the model Don t retain a discredited model Don t fall in love with your model Golomb S. W., Mathematical models uses and limitations. Simulation 4 (14), , 1970

25 Summary Definitions Validation System analysis (Hypothesis & abstract) Simulation Modeling

26 Course Outline Mechanistic Models Differential equations Variational Principle Solution to ODEs Classical ODE systems Cellular automata Others: inverse problems, difference equation, etc Phenomenological Models Regression Neural networks Others: power law, networks, etc