Research & AlgorithmsDANIEL SCHMITTER · RESEARCH & ENGINEERING

Scientific ML · Research & Algorithms

Build boundary conditions into a learned model instead of penalizing them

A learned trajectory should not have to negotiate its starting position and velocity with a loss function. Some constraints can be part of the coordinates themselves.

Daniel Schmitter · Research story · Revised draft

Read the paper ↗Paper PDF
EXPLORE THE IDEA

Change the path, keep the boundary

An editable waypoint with fixed endpoint positions and tangents.

CONSTRAINED TRAJECTORYMove the waypoint, not the endpointsFIXED BOUNDARY DATAPosition and tangent stay attachedLEFTy = 0.20y′ = 0.36RIGHTy = 0.40y′ = −0.24Interior y = 0.50Two C¹-connected cubic Hermite cells
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Two computed cubic Hermite cells share an interior value and derivative. The slider changes that value; both outer values and outer derivatives remain fixed. This illustrates a parameterization, not an optimized physical trajectory.

Follow the information

From input to outcome

The decoder combines free coordinates with a particular feasible solution. Since BN = 0 and Bc₀ equals the prescribed boundary data, changing z cannot violate those linear constraints. A fully constrained single cubic has no remaining free coordinates.

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

Free coordinates z → Constraint decoder → Hermite / spline field → Constrained output. The decoder combines free coordinates with a particular feasible solution. Since BN = 0 and Bc₀ equals the prescribed boundary data, changing z cannot violate those linear constraints. A fully constrained single cubic has no remaining free coordinates.
Information-flow map. An explanatory parameterization, not a new constrained-training benchmark. 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.

A boundary is sometimes a fact, not a preference

Imagine learning a short motion that must begin and end at prescribed positions and velocities. A penalty tells the optimizer that violations are expensive. A coordinate construction can make those particular violations impossible within the represented family. The difference is valuable when a constraint is known exactly and should not trade against another objective.

A constrained output layer, not an endpoint penalty

A one-dimensional physical response may need to meet a prescribed value and tangent at each end of an interval. Cubic Hermite coordinates put those four quantities directly in the representation. The derivative coordinates are multiplied by the interval width so that the physical slope does not change when the coordinate is normalized.

For a larger spline component, the same architectural idea writes coefficients as a particular feasible vector plus directions in the null space of the boundary constraints. A network can predict or optimize only those free coordinates. The output layer then maps them into a function that satisfies the linear boundary conditions at every update.

The information and computation flow of this example
Explicit component and information boundaries. Original scientific diagram; the stated component and information flow, not an additional experiment. Open full-size figure ↗

Give the endpoints their own coordinates

A cubic Hermite segment uses its two endpoint values and two endpoint derivatives. Fixing those coordinates enforces the corresponding conditions. A single cubic with all four fixed has no remaining freedom; several connected segments introduce interior degrees of freedom. The animation changes one interior value, not the supposedly fixed endpoints.

The physical units matter

The tangent basis multiplies a physical slope by the cell width. Forgetting that width changes the derivative when the grid changes. This small detail connects geometry to scientific ML: a parameter’s meaning must survive rescaling, refinement, and transfer. Exponential Hermite forms can additionally reproduce supplied dynamical modes, but require their own admissibility and conditioning analysis.

f(x)=H00(t)y0+hH10(t)m0+H01(t)y1+hH11(t)m1,t=(x−x0)/h\begin{gathered}f(x)=H_{00}(t)y_0+hH_{10}(t)m_0+H_{01}(t)y_1+hH_{11}(t)m_1,\quad t=(x-x_0)/h\end{gathered}
The factors of h convert physical slopes into normalized-cell coordinates. Two connected cells can keep outer constraints fixed while their interior changes.

What the construction does not promise

Large tangents do not provide arbitrary resolution inside a coarse cell. Exact endpoints do not prevent overshoot, preserve monotonicity, or satisfy every physical inequality between knots. Those are additional constraints or validation questions. The value of Hermite coordinates is a precise guarantee about a specific boundary—not a universal solution to motion learning.

Choose which facts the learner cannot violate

This is attractive when a boundary is known exactly, such as a clamped displacement or a prescribed endpoint value. It is less appropriate to freeze a noisy measurement as a hard truth. The construction is classical and this article is an explanatory component of the growth paper, not a new benchmark of constrained PINN training. Its value is the explicit separation between facts encoded in an output layer and quantities that remain learnable.

A component for constrained learning

A network can predict the free interior coordinates while a deterministic decoder enforces the known endpoint data. Training then addresses what is unknown. The companion manuscript places this parameterization next to exact refinement and explains which operations preserve the represented function. The guarantee comes from the decoder’s mathematics; an application benefit still needs a task-level experiment.

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. Operator-spline theory: consolidated research manuscript. Daniel Schmitter (2026). Local archive snapshot.
  2. Cardinal Exponential Splines: Part I—Theory and Filtering Algorithms. Michael Unser and Thierry Blu (2005). Primary literature.
  3. Consolidated research results, including constitutive edges and continual memory. Daniel Schmitter (2026). Local archive snapshot.

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