Research & AlgorithmsDANIEL SCHMITTER · RESEARCH & ENGINEERING

Scientific ML · Research & Algorithms

Learning physical laws from noisy data without differentiating the noise

Noisy measurements become especially troublesome when learning requires derivatives. A weak formulation changes the question: compare integrated physical effects instead of differentiating each imperfect sample.

Daniel Schmitter · Research story · Revised draft

Read the paper ↗Paper PDF
EXPLORE THE IDEA

Integrate through the noise

A local window asks a better-conditioned question.

OBSERVATIONSA window moves across noisy samplesAN INTEGRATED QUESTIONWeighted average in the window-0.332from 24 contributing samples∑ wᵢ yᵢ / ∑ wᵢSynthetic observations; no fitted uncertainty
50%
Deterministic synthetic noisy samples and a compact cosine test window. The readout is their normalized weighted average, not an estimated physical law or a denoising-accuracy result.

Follow the information

From input to outcome

Known test functions act on the observations before fitting. Their discrete adjoint defines the derivative transfer, so the learner need not fit naive pointwise derivatives of noise. Overlapping weak rows are not independent experiments.

Scroll the diagram horizontally to follow the route. Keyboard: focus the diagram, then use the arrow keys.

Noisy field samples → Weak observation map → Weak regression rows → Fit flux coefficients → Rollout prediction. Known test functions act on the observations before fitting. Their discrete adjoint defines the derivative transfer, so the learner need not fit naive pointwise derivatives of noise. Overlapping weak rows are not independent experiments.
Information-flow map. At the higher tested noise level, the simpler weak polynomial wins. Original vector schematic based on the method and evidence discussed in this article; signal shapes and icons are illustrative, not additional measurements. Open full-size diagram ↗

Read the main route from left to right; labelled side branches show additional inputs, checks or feedback. The sections below explain the operations and their experimental limits.

The derivative amplifies the wrong thing

A measured field contains physical variation and observation noise. Subtracting neighboring samples and dividing by a short interval can greatly amplify the latter. If those derivatives become targets, a law learner may explain measurement artifacts. Denser sampling does not automatically solve this; it can make a naive derivative estimate more sensitive. The learning problem begins before the choice of model.

Change the observation layer before changing the network

The trainable coefficients need not change when noisy measurements become difficult. What changes is the equation that connects observations to those coefficients. Pointwise fitting estimates time and space derivatives of a noisy trajectory. Weak fitting combines measurements over a window and moves derivatives onto known test functions. Both ultimately feed a small regression system.

The implementation must respect its discrete operators. A sampled continuous test derivative is not necessarily the adjoint of the numerical difference used on the data. The archived construction uses that discrete adjoint explicitly. Its weak rows still overlap in time and space; thousands of rows are not thousands of independent experiments.

Learn the law inside the simulator
Learn the law inside the simulator. Original scientific diagram; the stated component and information flow, not an additional experiment. Open full-size figure ↗

Move the derivative to something we control

Multiply the physical equation by a smooth test function and integrate. Integration by parts transfers derivatives from uncertain measurements onto the known test function, with boundary terms handled explicitly. The weak equation asks whether a candidate law explains weighted behavior over a space–time region. This classical principle is used in weak system identification; it is not a new spline invention.

∫utψ dt=−∫uψt dtif ψ vanishes at the endpoints\begin{gathered}\int u_t\psi\,dt=-\int u\psi_t\,dt\quad\text{if }\psi\text{ vanishes at the endpoints}\end{gathered}
Data remain under an integral. The derivative acts on a test function whose behavior we know.

The discrete equation matters too

Our experiment uses compact temporal windows and spatial sine/cosine tests. A subtle detail matters: the temporal operator uses the adjoint of the discrete difference. Sampling a continuous derivative does not necessarily preserve the discrete integration-by-parts identity. Exact calculus is valuable only if the observation model and its numerical implementation describe the same calculation.

A useful improvement, and a useful crossover

At 1% observation noise, median amplitude-shift rollout error falls from 0.03844 for pointwise dictionary fitting to 0.001401 for weak fitting. A weak polynomial gives 0.003194. At 2% noise, the weak dictionary still beats its pointwise counterpart, but the simpler weak polynomial is better: 0.005207 versus 0.006581. Richer representation and better observation handling are different sources of benefit.

Integration is not free information

A test window can suppress noise while hiding fine-scale physics. Its support, boundary behavior, and frequency content determine what is identifiable. Our three-seed synthetic study supplies the outer dynamics and generates controlled noise. It does not reproduce missing variables, irregular sensing, or instrument artifacts from a laboratory. Those uncertainties need their own observation model.

Better evidence before a bigger model

For scientific ML, this suggests an observation layer that expresses trustworthy integral relations before a learned component is fitted. It may improve the information presented to a simple learner more than adding capacity would. The higher-noise polynomial win is important: better observation geometry does not imply that the richest dictionary is the best estimator.

Change the evidence before enlarging the model

A noisy pointwise derivative may be the wrong learning target even for a powerful architecture. The operator view encourages a different representation of evidence: known tests, explicit boundary terms, and coefficient-domain calculations. Splines can support that machinery, but the demonstrated robustness here arises primarily from weak observation formation. The strongest model is not always the best first intervention.

Evidence & further reading

The links below distinguish the project record from foundational literature. This revised story does not add a new application-validation experiment.

  1. Consolidated research results, including constitutive edges and continual memory. Daniel Schmitter (2026). Local archive snapshot.
  2. Operator-spline theory: consolidated research manuscript. Daniel Schmitter (2026). Local archive snapshot.

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