Give your next idea
a mathematical foundation.

Higher-order physics. A trainable residual.

Directional Taylor jets provide derivatives along chosen directions. Mixed jets represent mixed partials. Direct Laplacian paths avoid constructing a Hessian when you only need its trace.

Start with a fourth-order residual, verify its boundary conditions, and retain gradients into the network parameters. A derivative demonstration alone does not establish PDE convergence.

Let the model learn its regions.

Soft partitions blend regional experts using nonnegative weights that sum to one. Thresholds, split directions, and expert parameters can train together.

Temperature hardening sharpens routing. It does not differentiate a pre-existing hard discontinuity. Select the partition package with its current AGPL-or-commercial license.

Ask for the operator you need.

Reuse field state to evaluate gradients, divergence, curl, Hessians, and supported contractions. Keep the backend, shape conventions, and derivative representation explicit.

For a deep Laplacian, choose the direct contraction rather than materializing every mixed partial. For mixed derivatives, account for the combinatorial coefficient count.

Match the curvature to the loss.

Use residual Jacobians, Hessian-vector products, and structured approximations on supported objectives. Gauss–Newton curvature and the full loss Hessian are distinct objects.

Start with a small objective and measure convergence, numerical error, and end-to-end runtime against your existing optimizer. No method wins on every task.

Make the assumptions visible.

Intervals, Taylor models, and checked certificate objects can bound supported quantities. The model, domain, and assumptions travel with the result.

Treat formal-kernel verification as a separately checked status. A sampled evaluation or a serialized certificate is not itself a theorem-prover result.

Evaluate your workload.
Sustain the shared tools.

Bring an equation, model, backend, and accuracy target. We can discuss a scoped evaluation or integration with independent numerical checks and a measured baseline.

Corporate partnerships fund public maintenance and examples. Technical implementation and support are scoped separately.