Why Domain Knowledge, Methodology, and Math Language Are Key to Good Models

Mathematical models serve as bridges between science and reality, quietly influencing everyday life—from weather forecasts and navigation to online shopping recommendations.

Large language models like ChatGPT are powerful mathematical models that convert natural language into vector representations and generate context‑aware responses through probabilistic inference.

Scenario: Electricity Consumption Forecast

Given China’s per‑capita electricity data from 1978 to 2023, how can we predict future values such as for 2030?

AI might answer by fitting the historical series with polynomial, exponential, or logistic functions, but this approach ignores external factors like economic trends, technology advances, and population changes.

A good mathematical model = domain knowledge + methodology + mathematical language

Thus, a more reasonable forecast should combine time‑series analysis with other important variables, forming a multivariate model.

Importance of Domain Knowledge

Without domain knowledge, even the most sophisticated mathematical techniques or AI algorithms become empty shells. For example, AI can excel at medical image analysis, but without expert guidance it may misinterpret noise as pathology.

Domain knowledge sets the starting point and direction of a model, while methodology and mathematical language provide the tools to express and solve the problem.

Does a More Complex Model Mean a Better Model?

Complexity does not guarantee quality. While detailed models can improve accuracy, they also demand more data, higher computational costs, and reduce interpretability.

Transparency is crucial: black‑box models raise trust and debugging issues, whereas white‑box models allow users to understand why a result is produced and to locate errors when they occur.

Why Build Models Manually When AI Is So Powerful?

AI cannot explain “why” it arrives at a result; it merely predicts based on patterns in data. Human‑crafted models, grounded in domain knowledge, provide explanations and can handle novel or extreme scenarios where data are scarce.

Understanding why a model works is more important than merely obtaining a result.

Human modeling combines intuition, creativity, and the ability to create new solutions beyond the limits of existing data.

Domain Knowledge + Methodology + Mathematical Language

The formula "Mathematical model = domain knowledge + methodology + mathematical language" stresses that an excellent model needs deep insight into the problem, a rigorous solution approach, and a clear expressive form.

We must ask ourselves:

"Do I truly understand the problem?"

"Is my model serving the problem or just showcasing technology?"

"Can my model clearly explain the underlying logic?"

Only by reflecting on these questions can we build meaningful models in an era dominated by AI.