For understanding uncertain and/or complex phenomena that are difficult to describe from the first principle, it is particularly important to appropriately formulate them as mathematical problems. This task is often termed "modeling". This course shows elemental mathematical techniques that are necessary for the modeling, illustrating representative probabilistic and/or nonlinear phenomena.
The goal is to acquire basic skills for engaging in advanced modeling of more complex phenomena by learning elemental mathematical models for probabilistic and/or nolinear phenomena.
Random variable, probability distribution, statistic, diffusion phenomena, Brownian motion, branching process, aggregation process, nonlinear phenomena, neural network model
|✔ Specialist skills||Intercultural skills||Communication skills||✔ Critical thinking skills||✔ Practical and/or problem-solving skills|
For each topic, we first show concrete target phenomena, and introduce how they are formulated as mathematical problems.
|Course schedule||Required learning|
|Class 1||Introduction to modeling||General introduction to observation, modeling, analysis and evaluation|
|Class 2||Observation of phenomena 1||Basic distributions and corresponding mathematical models|
|Class 3||Observation of phenomena 2||Basic statistical quantities and corresponding mathematical models|
|Class 4||Observation of phenomena 3||Stationarity, reversibility and irreversibility|
|Class 5||Modeling of diffusion phenomena 1||Macroscopic irreversibility of diffusion phenomenon, and derivation of diffusion equation|
|Class 6||Modeling of diffusion phenomena 2||Microscopic view of diffusion phenomenon and Brownian motion|
|Class 7||Modeling of diffusion phenomena 3||Markov process and mathematical description of Brownian motion|
|Class 8||Modeling of diffusion phenomena 4||Examples of diffusion phenomena|
|Class 9||Modeling of branching and aggregation phenomena 1||Branching process and its modeling|
|Class 10||Modeling of branching and aggregation phenomena2||Aggregation process and its modeling|
|Class 11||Modeling of nonlinear phenomena 1||Description by ordinary differential equation. Linear stability analysis around fixed points.|
|Class 12||Modeling of nonlinear phenomena 2||Nonlinear oscillators and Hopf bifurcation|
|Class 13||Modeling of nonlinear phenomena 3||Coupled oscillators. Understanding of mechanism of synchronization using coupled phase oscillators.|
|Class 14||Modeling of neural networks 1||Classification of neural network models. Mathematical characterization of abilities and limitations of perceptron.|
|Class 15||Modeling of neural networks 2||Auto associative memory model. Understanding of memory retrieval using the notion of Lyapunov function.|
To be distributed electronically when needed.
Students' understanding will be assessed by final exam.
Basic knowledge and skills about linear algebra, calculus, probability theory, and statistics are required.