Physics

In nefi the physics is a hard constraint: every candidate field is pushed through a differentiable Operator, and the prediction is exactly the discretized physics applied to it. Every operator evaluates at any curriculum resolution (at_resolution), runs on CPU and CUDA, declares its homogeneity when it has one, and carries a fidelity tag. Every benchmark measurement is simulated by a different, higher-fidelity model — a finer grid, float64, an independent scheme — so that no result relies on an inverse crime.

The families

family	forward model	gradients	instances	benchmark data from
Convolution fft_convolution	y = k ∗ x, N-D, zero-padded FFT, kernel rebuilt per resolution	autograd · batchable	toy1d, deconvolution, deconvolution3d	PSF on a 2× grid, float64
NV dipolar spectra nv_relaxometry, nv_direct	tensor power kernel × Lorentzian (F2), coherent F1, source-side direct sum F3	autograd · batchable (F1/F2)	nv_relaxometry	F3 direct sum, float64
Planar magnetostatics current_density, magnetization, biot_savart	stray field of sheet currents / magnetization at a standoff	autograd · batchable	current_density	real-space Biot–Savart sum, float64
X-ray transform radon, radon3d	parallel-beam line integrals, all views in one grid_sample	exact autograd adjoint · batchable	sparse_view_ct, ct3d	2× finer image grid, float64
Poisson poisson	−κΔu = f, spectral DST solver	autograd · batchable	poisson_source	finite differences on a 2× grid, float64
Variable-coefficient elliptic elliptic, eit, darcy, diffuse_optical	−∇·(σ∇u) + κu = f, harmonic-mean finite volumes, Jacobi-PCG	implicit-function-theorem adjoint, warm starts	eit, darcy_flow, dot3d	finite volumes on a 2× grid, float64
Transient heat heat (heat_explicit for data)	implicit Euler, harmonic-mean faces, Jacobi / Chebyshev / CG sweeps	discrete adjoint at trajectory memory	thermal_tomography	explicit substepped simulator, float64
Acoustic waves wave, wave_initial_condition	leapfrog, 4th-order Laplacian, PML (2-D) or sponge (3-D)	autograd or block checkpointing	wave_fwi, photoacoustic3d	2× finer grid, float64
Scattering born, lippmann_schwinger	Born / Rytov; Lippmann–Schwinger for the data	autograd · batchable (Born)	diffraction_tomography	iterated multiple scattering on a 2× grid, float64
Coherent optics holography, phase_object	angular-spectrum propagation of a thin phase object	autograd · batchable	holography	2× sampling, float64
Reaction–diffusion gray_scott	Gray–Scott, explicit Euler hitting the snapshot times exactly	autograd or block checkpointing	reaction_diffusion	4× smaller time step, float64
Bring your own fourier_sampling, phase_retrieval, beer_lambert, sampling, saturation, time_stepper, function	Fourier sampling (MRI), phase retrieval, Beer–Lambert, sampling, saturation, generic time stepping, any function	autograd; checkpointed time stepping	your problem (BYOP)	your generator

nefi list operators prints the full registry (36 entries, including the wrappers nuisance, stack, sum, pointwise and identity). Batchable operators can solve many measurements in one loop with nefi.batch_invert.

Pages

• Elliptic & magnetostatics

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  One differentiable steady-state solver with an exact implicit-function-theorem adjoint behind EIT, Darcy flow and diffuse optics; planar magnetostatics for NV current imaging; a new elliptic instance in about 30 lines.

• Wave, scattering, optics & reaction

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  Time stepping and its memory trade-off, acoustic waves with PML, Born and Lippmann–Schwinger scattering, angular-spectrum optics, Gray–Scott dynamics.

• Heat equation (NeFTY)

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  The implicit-Euler finite-volume solver, its discrete adjoint at trajectory-sized memory, the inner solvers and the explicit data simulator.

• Write your own

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  An operator is a differentiable function with at_resolution and a fidelity tag; pair it with an independent data generator and every tool in nefi works with it.