Why can’t an artificial neural network figure out what a lava lamp will do next?

In principle, if it's all "just physics" then you don't need machine learning at all - you can just simulate it to an arbitrary level of precision in deterministic code. The problem though is that "chaotic systems" are, well, "chaotic". Meaning, among other things, they are so complicated, with some any interactions among simultaneously changing elements, and so much sensitive to initial conditions, that it's effectively impossible to simulate them to any high degree of precision. This is the same reason why accurate weather forecasting has been such a challenge for so long, and only in the last couple of decades have we gotten kindof good at it. And even then it takes massive amounts of compute (think, supercomputers) to do that.

So if you had a good physics simulation package, and a supercomputer with an unlimited budget, you almost certainly could model your lava lamp pretty well.

Machine Learning can help but it does require, as you note, patterns in the underlying behavior. And it's those patterns that are hard to suss out. It's probably easier to use ML to get some vaguely useful approximation, than to use a deterministic physics simulation package to get a useful answer, but in both cases you're up against the same constraints: you'd need massive amounts of compute, incredibly high precision measurements, and lots of time.

It can predict it to an extent; however the problem (lava lamp) has strong sensitivity to initial conditions. To predict what happens after some time, the input must be measured so exactly that it is not possible to measure it precisely enough to predict its future state. You can't measure anything 100%; you hit a physical limit as some point.

See https://en.wikipedia.org/wiki/Double_pendulum

The purpose of data driven methods like training a model is that we want to be able to predict an output from a particular given input,

$$y = f(x).$$

The problem is that many relationships and dependencies cannot properly be captured by a simple model $f$. Thus, we introduce millions (or even billions) of parameters that we then tune. Tuning these parameters mean that we do computations on training data, and store these numbers somewhere. In the final step, we get a model $f$ that can predict an outcome or classify novel input data.

But if the relationships are very complex (as in the case of a lava lamp, which in most cases is consider a chaotic system), not even a very complex model $f$ can predict an outcome accurately.

You could of course model and simulate a lava lamp, but machine learning tries to abstract all that away and just look at the data. The central idea in machine learning is that the training examples are drawn from an underlying, typically unknown probability distribution that is assumed to represent the overall space of possible occurrences. The objective is then to construct a generalizable model of this space, enabling it to generate accurate predictions for previously unseen instances.

On one extreme you have empirical models (these are the typical AI models, like decision trees or LLMs). Empirical models are mathematical representations based on observed data and experiments rather than theoretical principles. They are created by fitting a mathematical equation to collected datasets, often using regression techniques, to predict outcomes for specific situations.

The other extreme are physical/mechanistic models. A mechanistic model is a mathematical or computational representation of a system. These models use equations and simulations to represent how a system works. Unlike empirical models that focus solely on data patterns, mechanistic models are built on established physical laws and mechanisms. Simulations are mechanistic models.

An outline of the methodological differences are shown in this picture (fig. 1 in this paper), as seen from biologists trying to model parasitic infections:

In reality, all models are on a continuous spectrum between these two extremes. Which approach you choose depends on what you are trying to do, and both approaches have their pros and cons.

This paper, with the figure below, summarizes it well (they discuss various models in the context of biological systems):

A predictive model might be selected for forecasting future process performance, while descriptive models might offer a greater understanding of relationships within a process (Tsopanoglou and Jiménez del Val, 2021; Sansana et al., 2021). Data availability is also a key concern; depending on the model type, there are limitations based on the quality and quantity of the data. For example, empirical models (Fig. 3) are best suited to processes with a large amount of experimental or observation data, for which mechanistic models consistently fail to predict outcomes; conversely, mechanistic models are derived from theoretical principles and are preferred when the data is lacking, coupled with a strong understanding of the mechanisms of the process, which quantitatively matches well reality (Tsopanoglou and Jiménez del Val, 2021; Sansana et al., 2021). In practice, every model is somewhere on the spectrum between purely empirical and mechanistic.

Returning to your original question, why can't an artificial neural network figure out what a lava lamp will do next?

Your lava lamp example is a case when empirical-faithful models tend to perform very poorly. This is because the interactions are very complex and hard to distill just looking at data. There is no "simple rule" for a machine learning model to learn.