Mathematical modeling

Mathematical modeling represents real systems with mathematical structures so their behavior can be analyzed, explained, or predicted.

Model formulation: The process of translating a real-world question and its assumptions into a mathematical representation. This is where a modeling problem becomes a defined system of variables and relationships.
Dynamical system: A mathematical description of how a system’s state changes over time. Many models represent evolving systems by specifying states and rules of change.
Weather forecasting: The prediction of future atmospheric conditions using observations and mathematical models. Numerical atmospheric models turn physical laws and measurements into forecasts.
Statistical model: A mathematical description of relationships among data variables that supports inference or prediction. Statistical models emphasize patterns in observed data, while mechanistic models encode proposed processes.
Overfitting: A modeling failure in which a model captures noise in its data and generalizes poorly. A model can match its calibration data closely yet make unreliable predictions elsewhere.
Parameter estimation: The process of inferring model parameters from observations or experimental data. Estimated parameters connect a model’s abstract quantities to measurements of the represented system.
Differential equation: An equation relating a function to one or more of its derivatives. Differential equations encode continuous change in physical, biological, and other systems.
Epidemiological modeling: The mathematical study of how diseases spread and affect populations. Compartment and network models estimate epidemic trajectories and compare interventions.
Mechanistic model: A model built from explicit representations of the processes that generate a system’s behavior. Mechanistic modeling is one strategy for grounding mathematical relationships in causal processes.
Model uncertainty: Uncertainty arising from imperfect knowledge of a model’s structure or assumptions. Even precise data cannot eliminate uncertainty caused by an incomplete representation of the system.

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