fix(integrators/fde): CaputoEuler.reset state desync + set_default_fdeint

[chaoming0625]

Fresh review of brainpy/integrators fractional/partial integrators.

• High — CaputoEuler.reset replaced its f_states Variables with plain Arrays → state desync → wrong results after reset under IntegratorRunner (0.911 vs correct 0.380). Now stored in a bm.VarDict so reset updates in place.
• Medium — fdeint(method='l1') literal default made set_default_fdeint a no-op for default-method calls; default method=None so the global fallback applies.

Core solver numerics (CaputoEuler/L1/GL) validated against reference recurrences; prior-audit FDE bugs (C-08, H-30/31/32) verified already-fixed. In-scope: 5 passed. Findings: docs/issues-found-20260619-integrators-fde-pde.md.

Summary by Sourcery

Fix fractional integrator state handling and default method selection for FDE solvers.

Bug Fixes:

• Preserve CaputoEuler memory state variables across reset calls to avoid desynchronization when used with IntegratorRunner.
• Ensure fdeint honors the globally configured default FDE method by changing the method parameter’s default to use the global fallback.

Documentation:

• Document the FDE/PDE integrator review, including validated numerics and previously reported issues, in a new issues-found report.

Tests:

• Add regression tests verifying CaputoEuler.reset keeps memory buffers as Variables and produces the same results as a fresh integrator run.
• Add a generic-level test confirming fdeint respects set_default_fdeint when the method argument is omitted.

[sourcery-ai]

Reviewer's Guide

Fixes a state desynchronization bug in the CaputoEuler fractional integrator reset path and restores correct honoring of the global default FDE method in fdeint, with regression tests and an audit report documenting FDE/PDE integrator issues and prior fixes.

Sequence diagram for CaputoEuler.reset state synchronization

sequenceDiagram
  actor User
  participant CaputoEuler
  participant VarDict as bm.VarDict
  participant Variable as bm.Variable
  participant IntegratorRunner

  User->>CaputoEuler: __init__(alpha, num_memory, inits)
  CaputoEuler->>VarDict: create VarDict({v: Variable(zeros)})
  VarDict-->>CaputoEuler: f_states (VarDict of Variable)

  User->>CaputoEuler: reset(inits)
  CaputoEuler->>VarDict: f_states[key] = bm.zeros(...)
  VarDict->>Variable: update .value in place

  User->>IntegratorRunner: IntegratorRunner(CaputoEuler, dt, inits)
  User->>IntegratorRunner: run(t_end)
  IntegratorRunner->>CaputoEuler: _integral_func(...)
  CaputoEuler->>Variable: read/write memory buffer

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Sequence diagram for fdeint default method selection via set_default_fdeint

sequenceDiagram
  actor User
  participant Generic as generic_module
  participant Global as _DEFAULT_FDE_METHOD
  participant FDEInt as fdeint
  participant CaputoEuler
  participant CaputoL1Schema

  User->>Generic: set_default_fdeint("euler")
  Generic->>Global: set _DEFAULT_FDE_METHOD = "euler"

  User->>FDEInt: fdeint(alpha, num_memory, inits, method=None)
  FDEInt->>Global: read _DEFAULT_FDE_METHOD
  FDEInt-->>FDEInt: method = _DEFAULT_FDE_METHOD
  alt method == "euler"
    FDEInt->>CaputoEuler: construct CaputoEuler(...)
  else method == "l1"
    FDEInt->>CaputoL1Schema: construct CaputoL1Schema(...)
  end
  FDEInt-->>User: return chosen integrator

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File-Level Changes

Change	Details	Files
Ensure CaputoEuler.f_states remains composed of registered bm.Variable entries across resets so IntegratorRunner and JAX state tracking stay consistent.	Initialize f_states as a bm.VarDict of bm.Variable buffers instead of a plain dict, removing the need for register_implicit_vars on the mapping itself. Update reset() to assign new zeroed buffers through the VarDict so underlying Variable values are updated in place while preserving Variable identity. Document the VarDict semantics in a reset() comment to clarify why in-place updates are required for JAX/stateful usage.	brainpy/integrators/fde/Caputo.py
Make fdeint correctly use the globally configured default FDE method when the method argument is omitted and add coverage for both the reset bug and default-method behavior.	Change fdeint signature so method defaults to None and dt is typed as an optional float, allowing the internal _DEFAULT_FDE_METHOD fallback to take effect. Add tests verifying that CaputoEuler.reset preserves Variable-typed memory and that a reset-and-run trajectory matches a fresh integrator run under IntegratorRunner. Add a generic-level test confirming that set_default_fdeint is respected when method is omitted and that different defaults select CaputoEuler vs CaputoL1Schema. Add an issues-found markdown report documenting validated numerics, fixed historical bugs, and remaining style-level findings for FDE/PDE integrators.	brainpy/integrators/fde/generic.py brainpy/integrators/fde/Caputo_test.py docs/issues-found-20260619-integrators-fde-pde.md

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