Many field models ask for a Laplacian: the sum of diagonal entries of a spatial Hessian. Materializing every Hessian entry is one valid way to obtain that sum, but it is not always the representation best matched to the requested quantity.
A complete field-equation example
Our PyTorch notebook solves a two-dimensional Poisson problem on the unit square. The manufactured solution sin(πx)sin(πy) supplies a known forcing and a reference field. A tanh network represents the solution, a direct omnibias Laplacian supplies the interior residual, and a separate penalty handles zero boundary values.
The code verifies the contraction against a dense Hessian trace at a few small probe points. During training it uses the direct contraction and checks gradients into the network parameters. After training it reports field error, boundary error, and normalized residual error separately.
Different evidence answers different questions
A low training loss does not establish small boundary error, and an operator microbenchmark does not establish a faster complete PDE solve. The worked example therefore publishes its source, architecture, sampling, optimizer, numerical checks, and output arrays. It makes no end-to-end speedup claim.
The result is also scoped to a smooth, small, known problem. Dimension, network width, conditioning, dtype, and derivative representation affect the workload. Direct contractions avoid constructing quantities that are not needed, but still require computation.
From a field equation to an engineering workflow
Poisson-type formulations provide a useful starting point for potential fields, steady diffusion, and research into simulation surrogates. Applying a method to a real device requires domain geometry, coefficients, boundary data, reference solvers or measurements, and an error budget.
Derivon helps make those choices explicit. Our work can include operator selection, differentiable model integration, reference calculations, and evaluation criteria for scientific machine learning. For inverse problems, calibration and identifiability need their own validation rather than being inferred from a forward demonstration.
Reproduce the result
The recorded notebook reports relative solution error of 0.056%. Inspect the conditions and raw data before comparing it with another method.
Read and run the complete notebook · Discuss a scientific ML project