Differentiation and restart

Physical gradients

Midtrain differentiates through the physical intervals connecting corrected endpoints. The semi-implicit core supplies backward operators for its free-surface and vertical solves. Physical checkpointing recomputes integration chunks during backward while retaining the complete state required by each chunk.

The implementations are in cores/implicit_p0_fixed/autograd.py, strict_vertical.py, gradient_kernels.py and hybrid/sparse_physics.py. The Explicit core uses its PyTorch integration operators.

Regression coverage

tests/test_fixed_core_runtime.py compares checkpointed and direct gradients, checks the sea-surface adjoint and verifies complete-state restart. tests/test_strict_vertical.py checks vertical operator gradients and finite differences. Training restart tests additionally preserve model, optimizer, sampler, numerical state and random-number state.

Run the suite with python -m pytest after the development installation.