Come, let's meet.
We start from scratch. No prior knowledge required — only curiosity.
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Nesin Mathematics Village · Şirince
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17 September 2026 · Nesin Mathematics Village · Şirince
And why can even the most powerful AI not remove the limits of prediction?
In this talk
First number
In a numerical experiment, I started two double pendulums one trillionth of a radian apart. They had the same masses, lengths, gravity and governing equations — no random input was added.
A difference smaller than one ten-billionth of a degree. In this simulation, with the selected parameters and divergence criterion, the two trajectories became macroscopically different after about 17 seconds.
Here is the interesting part: as I reduced the initial separation by a factor of 1000 at each step, I measured the time required to reach the same divergence threshold:
I ran the same double pendulum twice, with a difference of only one trillionth (10⁻¹²) of a radian in the starting angle. At 4.3 seconds you cannot even see a difference. At 11.3 seconds it becomes visible. At 14.7 seconds the two pendulums point in completely different directions. At 17.2 seconds they have nothing in common — as if they had never started from the same place. Obtaining these four numbers took a real feat of computational engineering: as Δθ₀ shrank, measuring it demanded a proportionally finer-stepped, higher-order numerical method — an ordinary simulation step cannot capture the last two values. Chaos does not make the measurement impossible; but it makes you pay the price in genuine computational effort.
Why does each row cost more than the last? Because the same exponential growth applies to the method's own rounding error: to resolve an initial difference 1000× smaller, the numerical error must be pushed below it too — a finer step and a higher-order method. A coarse, low-order step misses the last two values; with a fine enough step and a high-order method (RK4 + Richardson) all four converge stably to the same number.
The underlying idea is that, along a chaotic trajectory with a positive Lyapunov exponent, sufficiently small perturbations grow approximately as Δ(t) ≈ Δ₀eλt over a finite interval. Solving this relation for the time to reach a fixed error threshold makes that time depend logarithmically on the initial uncertainty.
In simulation you can pay this cost in computing power and measure every value. But in real weather forecasting you cannot refine the initial conditions without limit — that is where the real barrier lies.
Lyapunov exponents quantify local rates of separation. They do not imply that every chaotic system diverges at one constant rate for all times, and weather uncertainty also includes observation, model and resolution errors.
One question, three acts
How do we step numbers forward in time — and what does that have to do with my computational electromagnetics and high-frequency simulation research?
Does knowing the rules exactly mean knowing the future? (This is exactly where the pendulum experiment lands.)
AI weather models such as GraphCast, Pangu-Weather, GenCast and ECMWF's AIFS have made forecasting much faster and, on many benchmarks, more accurate. But they have not removed the predictability limit — and we will see why.
The broader lesson is not that such systems are completely unknowable, but that sufficiently precise long-range trajectories may be unattainable even when the governing rules are well understood. In practice, prediction becomes probabilistic.
A short note on universality
The Feigenbaum constant whispers a useful idea: very different systems can share the same mathematical scaling when they approach chaos through a particular route.
In the talk, this appears only as a brief pause on universality. It does not mean that every chaotic system follows this route; atmospheric predictability has its own multiscale dynamics and is not derived from this constant.
Why I am giving this talk
Numerical modelling is not an abstract subject for me — it lies at the centre of my research on computational electromagnetics, high-frequency system simulation and finite-difference time-domain modelling (FDTD). Weather models and FDTD do not solve the same equations, but they share a core numerical idea: continuous physical laws are discretised in space and time so that a computer can evolve an approximate solution step by step.
Come, let's meet.
We start from scratch. No prior knowledge required — only curiosity.
Location QR
Scan with a phone to open the location search. The map link is ready for directions.