Mathematical modelling translates the behavior of real-world systems into mathematical equations. Broadly, two main approaches exist: empirical modelling, which fits equations directly to observed data, and phenomenological (or physical) modelling, which builds equations from fundamental scientific laws.

In practice, the most powerful solutions blend these approaches. Semi-empirical modelling does exactly this. It leverages the data-driven strength of empirical methods while incorporating known scientific principles about the process.

Nonlinear modelling is a specialised form of empirical or semi-empirical modelling. It moves beyond simple linear relationships to capture the more complex, curved interactions that are common in real-life systems, such as diminishing returns, threshold effects, and synergistic relationships.

A car's speed depends on several factors

Consider predicting a car's speed. While the accelerator position is primary, numerous other factors are at play: road friction, slope of the road, air resistance, fuel quality, and transmission losses. A purely physical model accounting for all these forces would be enormously complex.

A nonlinear modelling approach offers a more efficient path. By measuring key variables ‐ accelerator position, wind speed, wind direction, fuel properties, and pairing this data with actual speed readings, we can build a good predictive model. This model learns the complex interplay of all factors, providing accurate speed predictions under various conditions without needing to solve every underlying physical equation.

Imagine you're trying to bake the perfect chocolate cake. You know the main ingredients: flour, sugar, baking powder, cocoa powder, and the process variables: oven temperature and baking time. But the result isn't a simple sum of its parts.

A linear (or "simple") model would suggest: "If one teaspoon of baking powder makes the cake rise a little, then three teaspoons will make it rise exactly three times as much." We know this isn't true. Too much baking powder creates large holes, a bitter taste, and the cake can even collapse.

• Sugar & Browning: Sugar doesn't just add sweetness; it caramelises. A little more sugar makes the crust a perfect golden brown, but too much causes it to burn much faster ‐ a sudden, nonlinear change.
• Baking Time & Temperature: Baking for 30 minutes at 180 °C is perfect. Baking for 60 minutes at 90 °C won't give you the same cake ‐ it's a soggy disaster. The relationship isn't a straight line; there's a crucial zone, an operating window which will result in good cakes.
• Baking Powder & Rise: An extra half teaspoon can make the cake perfectly fluffy. But add a full extra teaspoon, and the cake rises too quickly, develops large tunnels, and gets a bitter, soapy aftertaste. The effect isn't gradual; it hits a "tipping point."

The quality of a chocolate cake depends on its composition and baking process variables

Instead of trying to write down every chemical reaction in the oven (the "phenomenological" approach), a nonlinear modeller would:

• Gather Data: Bake a dozen cakes, carefully changing one or two variables at a time (e.g., sugar amount, oven temperature, baking powder) and measuring the outcome (fluffiness, sweetness, browning).
• Find the Pattern: The modeller's algorithms would detect the hidden relationships: "Ah, I see. When sugar is above 250g and oven temperature is above 190 °C, the burn score increases dramatically ‐ that's a nonlinear interaction."
• Build the Predictive Model: Finally, he would create a "Cake Quality Predictor." You would specify your desired sweetness, fluffiness, and crust color, and using the nonlinear model, you could determine the exact recipe, baking time and temperature needed to achieve it, perfectly balancing all those complex, interacting factors.

Nonlinear modelling is the tool that helps us navigate and predict these complex, real-world behaviours when simple, straight-line thinking fails.

Modern methods of nonlinear modelling based on artificial neural networks usually provide superior accuracy and flexibility. The key to success lies in expertly combining these tools with domain-specific knowledge ‐ whether it's insights into materials, process nonlinearities, or elements of physical models ‐ to create reliable, interpretable solutions.

Nonlinear Solutions Oy specialises in this exact discipline. We possess the deep know-how, practical experience of decades, and an advanced toolset tuned over 30 years required to develop effective nonlinear models that solve real-world engineering and business problems, with our work frequently featured in industry publications and conferences. With our tools and know-how, a much smaller amount of data is needed. Many of our projects have been done with hardly 20 observations.