Physical modeling, simulation, and the overlap with machine learning.

Part 1 of a five-part series on digital twins with System Modeler and Wolfram Language. One winter-wheat field west of Linköping: Sentinel-2 reads its greenness every few days, SMHI supplies the weather, a stock-and-flow model predicts the season, and a Wolfram Databin keeps the numbers where any browser can reach them. Calibrated on five past seasons, running live on the one underway, every source free and public—and the twin already disagrees with the satellite, for a reason the model can explain.

The Swedish Air Force turned 100, so I flew the century in simulation: seven aircraft from one Modelica library, from a hands-off glider to a drone that flies its whole mission alone. Every era fixed the last one's limitation and bought a new one—the piston ran out of air at 9 000 m, the J 35-class delta reached 11 km in 7.2 minutes but burned 30% of its fuel doing it, and the battery brought back the glider's arithmetic. Simulated in System Modeler, animated in Blender.

The transformer on your street is a 140-year-old design: iron, copper and a tank of oil. I modeled a neighborhood in Modelica and threw the same short circuit at the old gear and its solid state replacement. The old breaker let 6 000 A run for a tenth of a second; the solid state transformer never passed 430 A. Simulated in System Modeler, rendered in Blender.

Got out to a ball that pitched on middle and hit off stump, so I modeled it—multibody flight, an aerodynamic side force, and a contact model for the bounce. Length turned out not to change how much it swings at all: 0.38 m every time. What length changes is how long you get to read it—81 ms off a full ball, 324 ms off a short one.

Built a Modelica stock-and-flow model of reader acquisition—tracking seven stocks, six feedback loops, and a three-month simulation. The model showed the warm pool of aware-but-unconverted readers was 20× the buyer count. That changed what I did next.

Three workflows using simulation as the training environment: a neural state predictor for EV battery temperature, a CTESN surrogate model for fast optimization, and a Q-learning agent trained entirely inside simulation. All grounded in physics, all traceable to equations.

Thermal model of a house in Linköping using 10 years of PVGIS climate data. Full energy demand and PV production simulation. Payback period: 16 years in Sweden, 6 in Delhi, 3.6 in Calama, Chile. Template notebook available for any location.

Continuous-Time Echo State Networks trained on simulation data to replace expensive physics models inside calibration and optimization loops. Matches full-model dynamics, runs orders of magnitude faster, and exports as a native System Modeler component.