Height is fitted to the plot. Normalized values divide by the sampled peak on x ∈ [−4, 4], y = 0: 0.7697.
BUILD WITHPyTorchJAXKeras 3
FORWARD SPATIAL DERIVATIVES. ORDINARY PARAMETER AUTODIFF.MEET THE IDEA ↓
01 / A DIFFERENT WAY THROUGH
Go deeper. Keep the gradient.
High-order physics asks more of a model than a first derivative. Nested spatial autodiff can grow expensive before training even begins.
omnibias carries supported derivatives forward through the network. Activation identities and Taylor jets compute the quantities you need while keeping gradients into the parameters you want to train.
Shared polynomial recurrences provide supported activation derivatives.
02
Carry the derivatives forward.
Compose directional jets or contract the specific operator your residual needs.
03
Train through the result.
Use the derivative in your loss. Ordinary parameter autodiff completes the loop.
02 / IDEAS YOU CAN TOUCH
Small changes. Different possibilities.
Explore the two limits behind derivative operators and trainable decisions.
TWO LIMITS. TWO DIFFERENT IDEAS.
Nearby shifts. An exact derivative.
Move the bias gap toward zero. The normalized difference approaches the first derivative. omnibias evaluates the analytic limit directly, avoiding subtraction of nearby samples.
[tanh(z + δ) − tanh(z)] / δ → tanh′(z)
Normalized finite difference┄ Analytic derivative
03 / FROM PRIMITIVE TO POSSIBILITY
Build beyond the first derivative.
Start with the quantity your model needs. Choose the mathematical path that fits.
The public omnibias substrate contains 16 Python distributions. Core derivatives and backend primitives use Apache-2.0; selected advanced engines offer AGPL or commercial licensing.
Application solvers live in separate repositories. Each package keeps its own scope, maturity, and license.
More of your physics. More of your model’s structure. Inside the training loop. Explore four workflows you can inspect, reproduce, and build on.
01 / MechanicsThe bottleneck
Drag to explore. Scrolling continues the story.
< 0.001%Relative solution error Recorded notebook run · CPU float64Inspect result ↗
01 / Mechanics
Put more physics inside the training loop.
Conventional workflow
A fourth-order residual asks a spatial autodiff graph to differentiate itself again and again.
With omnibias
Carry the required derivatives forward. Train through the residual with gradients into every model parameter.
< 0.001% relative solution error · recorded example
Derivon connects the physics, boundary construction, derivative representation, and independent error checks into an engineering workflow.
A smooth, nondimensional, manufactured beam problem. Exact boundary conditions come from its polynomial envelope; global PDE convergence is not implied.
A Laplacian needs a Hessian trace. Constructing every Hessian entry does extra work for that particular quantity.
With omnibias
Propagate the supported contraction directly, then use it in a trainable field equation.
0.056% relative solution error · recorded example
Derivon selects operator representations and validation criteria for simulation surrogates, field models, and inverse-problem research.
A unit-square Poisson problem with a known smooth solution. Boundary values are penalized, not exactly enforced. This is not a measured training-speed comparison.
Parameter fitting needs more than a loss function: parameterization, residual sensitivities, and solver assumptions determine the result.
With omnibias
Use residual-Jacobian curvature in a supported Gauss–Newton solver, keeping the parameter and residual contracts explicit.
< 1e-12 held-out rmse · recorded example
Derivon designs calibration workflows around identifiability, constraints, noise, and numerical conditioning before choosing an optimizer.
Three parameters and noiseless synthetic decay observations. Recovery at floating-point precision is not a robustness or uncertainty guarantee for noisy measurements.