The physics model that stays uncertain when the law is noiseless

Suppose a table contains time in one column and the exact displacement of a damped spring in another. There is no sensor noise. With enough points, the honest predictive distribution should collapse onto one curve—not merely get its mean closer while continuing to hedge.

Tenachi and colleagues test four tabular foundation models on 316 analytic physical equations. These models solve a new table in one forward pass, using patterns absorbed during pretraining as a prior. They are excellent interpolators and beat six trained baselines across almost every benchmark stratum. Then the noiseless limit exposes what that prior rules out.

A prior describes which data-generating mechanisms are possible before the current table is observed. The studied models were largely pretrained on synthetic structural causal models—random graphs whose nodes usually include independent noise. If every mechanism in that training universe is noisy, a truly deterministic law has no seat at the table.

The crucial diagnostic separates predictive width from root mean squared error (RMSE). When width plateaus while RMSE falls, calibration worsens even though point predictions improve. This tiny ratio is the arithmetic used by the interactive trace below.

def uncertainty_to_error(width, rmse):
    if width < 0 or rmse <= 0:
        raise ValueError("invalid uncertainty or error")
    return width / rmse

In this illustration the final width-to-error ratio is 1.4. The paper does not report that synthetic ratio; it reports the same qualitative separation across context sizes. At noise standard deviation 0.1, widths return to the expected calibration scale, which localizes the failure to the deterministic boundary rather than to the interval estimator generally.

Under both defaults and tuning, TabPFN-3 leads every reported stratum except one: at the largest context, the Gaussian process beats the foundation model's cheapest setting. Even out of domain at the largest context, its mean rank is 8.47 across 314 successfully scored tasks, versus 9.85 for the tuned and ensembled Gaussian process.

The authors feed 200 noiseless points from time 0 to 2 and query out to time 6. Every tested model relaxes to a constant outside the observed range, usually within about half a period. TabICLv2's density keeps wave-like bands, but bands can preserve amplitude without preserving phase.

The unshifted truth ranks only 98th for TabICLv2 and 157th for TabPFN-3. The density is genuinely multimodal, but it has not recovered the oscillator's frequency and phase. That negative control prevents an attractive figure from becoming a false mechanism claim.

Units provide another kind of structure. Buckingham's Pi theorem says that if a law has n variables and their SI-dimension matrix has rank r, only n − r independent dimensionless groups are needed. Five raw columns can therefore encode a problem with only three effective groups.

A column-permutation-invariant, unit-blind model sees only five columns. In the paper, error for every model tracks raw column count. More strikingly, TabPFN-3 becomes significantly worse as the number of Pi groups rises, while RealMLP and CatBoost become better. The trained methods can exploit a simplification that the strongest pretrained model treats as extra difficulty.

Tuning is not perfectly symmetric either. Trained baselines reserve 20% of each sample budget for validation and fit on 80%, whereas foundation models condition on every row. That handicap is largest at 50 samples. Meanwhile the uncertainty comparison includes only the Gaussian process among trained baselines, because the others emit point estimates.

• Test predictive width separately from point error at zero and nonzero noise.
• Probe phase or conservation laws, not merely wave-shaped output.
• Provide units and compare against dimensionless-group features.
• Use a true extrapolation shell so out-of-domain points cannot leak back inside the training box.
• Audit generating processes as well as dataset names when checking contamination.

The practical bridge is to the decision-tree and linear-regression pages: strong generic tabular learners remain valuable controls, but physics supplies invariances they will not discover from a column permutation alone. Before calling a posterior scientific, ask whether its hypothesis space includes the noiseless law you hope to learn.