Research & AlgorithmsDANIEL SCHMITTER · RESEARCH & ENGINEERING

Scientific ML · Research & Algorithms

Beyond a direct KAN: learning the physical law that extrapolates

Instead of asking a network to rediscover an entire dynamical system, learn the one response law the physics leaves unknown. A small model can then travel inside a much stronger solver.

Daniel Schmitter · Research story · Revised draft

Read the paper ↗Paper PDF
EXPLORE THE IDEA

Learn the missing law

Known transport and diffusion surround one unknown response.

A FIELD EVOLVESKnown conservation structureuₜ = νuₓₓ − ∂ₓF(u)LEARN THIS COMPONENTThe constitutive response FState range ±1.35Flux is computed; field is schematic
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The colored x–t field is an analytic schematic, not a PDE rollout. The flux plot evaluates F(u)=u²/2+0.08 sin(4u). The learned-versus-direct comparisons below are archived experiments; the flux dictionary contains this generating component.

Follow the information

From input to outcome

Learning identifies a scalar flux law; the supplied solver turns that law into a nonlinear evolution. The direct KAN and MLP controls instead learn the non-diffusive derivative. These paths receive different structural priors.

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

Observed trajectories → Identification rows → Fit the unknown law → Known PDE solver → New trajectory. Learning identifies a scalar flux law; the supplied solver turns that law into a nonlinear evolution. The direct KAN and MLP controls instead learn the non-diffusive derivative. These paths receive different structural priors.
Information-flow map. The matched dictionary contains the generating sine; local spline tails did not extrapolate well. 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.

Put learning where the uncertainty lives

Suppose we know that a field diffuses and that its transport conserves a quantity, but not how flux depends on local state. One approach learns the entire non-diffusive right-hand side. Another learns only that scalar response law and lets a known numerical solver perform the evolution. These are different uses of data—and they lead to different extrapolation mechanisms.

Two ways to spend the same learning effort

In the structured path, trajectory measurements fit a small scalar law. A known solver supplies conservation, diffusion, and time evolution. In the direct path, a 2–8–1 cardinal KAN or a 2–16–16–1 MLP predicts the non-diffusive state derivative from state and spatial gradient. Both paths still use known viscosity. They are therefore different placements of learning, not a contest in which every architecture receives exactly the same prior.

The distinction matters outside the training amplitude range. The learned law can be evaluated inside the same solver at a new state, but its behavior there is determined by its representation. A trigonometric dictionary containing the true oscillation has an unusually favorable continuation. A local spline tail has no corresponding promise.

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 ↗

A nonlinear system with a small learnable core

In the equation below, viscosity and the outer derivative structure are given. The unknown function is F. If F is expanded in fixed basis functions, its coefficients enter identification linearly, even though the resulting dynamics remain nonlinear. Learning can become a compact regression rather than a search over every state-to-derivative map. The hard questions shift to excitation, representation, and behavior outside the observed range.

ut=νuxx−∂xFθ(u)⏟learn this law\begin{gathered}u_t=\nu u_{xx}-\partial_x\underbrace{F_\theta(u)}_{\text{learn this law}}\end{gathered}
Known evolution structure surrounds a small unknown function. Rollout accuracy tests whether the learned function works inside that structure.

A striking result, with a precise reason

For the synthetic flux one-half u squared plus 0.08 sin(4u), a sparse polynomial–trigonometric library reaches median higher-amplitude, longer-horizon normalized error 9.29 × 10⁻⁷ across three seeds. The direct cardinal KAN control reaches 0.330 and the MLP 0.298. Crucially, the library contains the generating sine component, and the structured model receives the conservation factorization. This is not a pure architecture contest.

The more interesting test is outside the dictionary

A saturating log-cosh flux is not exactly contained in that finite library. The approach reaches median error 0.00520, versus 0.0411 for a structured degree-five polynomial and 0.1165 for the direct MLP. This suggests the representation can help beyond exact component recovery. It remains three deliberately constructed cases, not evidence that arbitrary unknown physics now extrapolates reliably.

Every seed in the two amplitude-shift panels. Lower is better. Matched oscillatory and unmatched saturating laws are separate, with all eight method families—not only the strongest comparison.
Every seed in the two amplitude-shift panels. Lower is better. Matched oscillatory and unmatched saturating laws are separate, with all eight method families—not only the strongest comparison.

Locality is not an extrapolation law

The local spline flux performs much worse under amplitude shift: median error 0.187 for the oscillatory case. Compact support controls where coefficients act, but unobserved tails do not become physically correct by being local. Adding a local residual to an accurate global model can also hurt. Choose a representation for how the unknown law should continue, not only how well it interpolates.

A simulator with a learnable physical core

The larger vision is a simulator that learns a missing material response instead of relearning all of mechanics. The near-exact matched-law result shows what is possible when the interface is right. The unmatched law and failed local tails tell us what still has to be established. That combination is a stronger ML story than a universal claim to beat KAN.

What “beyond a direct KAN” should mean

The experiment supports a more useful question than which network name wins: what must be learned, and what can the model already know? A physics-structured KAN would be a different comparator. The winning dictionary here is not an exponential B-spline kernel. The connection to the operator toolbox is the placement of learning and coherent global components, not a universal spline advantage.

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. KAN: Kolmogorov–Arnold Networks. Ziming Liu et al. (2024). Primary literature.
  3. Operator-spline theory: consolidated research manuscript. Daniel Schmitter (2026). Local archive snapshot.

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