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"""Backpropagation-free simulator for the Dillavou two-edge circuit.
The circuit equations and standard/overclamping updates follow Appendix D/F
of Dillavou et al. (arXiv:2505.22887v2). Every adaptive operation is an
explicit NumPy local rule; this module intentionally has no autodiff path.
"""
from __future__ import annotations
from dataclasses import dataclass
from typing import Callable, Iterable, Optional
import numpy as np
Array = np.ndarray
@dataclass(frozen=True)
class Circuit:
high: float = 0.4351
low: float = 0.0181
conductance_per_gate: float = 8.5e-4
threshold_voltage: float = 0.7
fixed_conductance: float = 1.0 / 500.0
measured_learning_rate: float = 2040.0
integration_step_seconds: float = 2.0e-4
gate_minimum: float = 1.0
gate_maximum: float = 5.2
@dataclass(frozen=True)
class Task:
name: str
input_voltage: float
label_voltage: float
@dataclass(frozen=True)
class LocalAffineBias:
reference_gate: Array
bias_at_reference: Array
local_slopes: Array
def __post_init__(self) -> None:
for value in (
self.reference_gate, self.bias_at_reference, self.local_slopes
):
if np.asarray(value).shape != (2,):
raise ValueError("two-edge bias arrays must have shape (2,)")
def __call__(self, gates: Array, strength: float = 1.0) -> Array:
gates = np.asarray(gates, dtype=float)
if gates.shape != (2,):
raise ValueError("gates must have shape (2,)")
return strength * (
self.bias_at_reference
+ self.local_slopes * (gates - self.reference_gate)
)
@dataclass
class LocalPredictor:
"""Independent per-edge affine filters trained by normalized LMS."""
reference_gate: Array
feature_scale: Array
coefficients: Array
affine: bool
@classmethod
def zeros(
cls,
reference_gate: Array,
feature_scale: Array,
*,
affine: bool,
) -> "LocalPredictor":
width = 2 if affine else 1
return cls(
reference_gate=np.asarray(reference_gate, dtype=float).copy(),
feature_scale=np.asarray(feature_scale, dtype=float).copy(),
coefficients=np.zeros((2, width), dtype=float),
affine=affine,
)
def features(self, gates: Array) -> Array:
gates = np.asarray(gates, dtype=float)
if gates.shape != (2,):
raise ValueError("gates must have shape (2,)")
if not self.affine:
return np.ones((2, 1), dtype=float)
normalized = (gates - self.reference_gate) / self.feature_scale
return np.column_stack((np.ones(2, dtype=float), normalized))
def predict(self, gates: Array) -> Array:
return np.sum(self.coefficients * self.features(gates), axis=1)
def update(self, gates: Array, neutral_measurement: Array, rate: float) -> Array:
"""One local normalized-LMS update and its pre-update residual."""
neutral_measurement = np.asarray(neutral_measurement, dtype=float)
if neutral_measurement.shape != (2,):
raise ValueError("neutral measurement must have shape (2,)")
features = self.features(gates)
residual = neutral_measurement - np.sum(
self.coefficients * features, axis=1)
normalization = np.sum(features * features, axis=1, keepdims=True)
self.coefficients += (
rate * residual[:, None] * features / np.maximum(normalization, 1e-12)
)
return residual
def copy(self) -> "LocalPredictor":
return LocalPredictor(
reference_gate=self.reference_gate.copy(),
feature_scale=self.feature_scale.copy(),
coefficients=self.coefficients.copy(),
affine=self.affine,
)
def free_output(circuit: Circuit, gates: Array, input_voltage: float) -> float:
"""Appendix D, Eq. D13, with gates ordered (minus, plus)."""
gate_minus, gate_plus = np.asarray(gates, dtype=float)
scale = circuit.conductance_per_gate
numerator = (
input_voltage * circuit.fixed_conductance
+ scale * (
gate_plus * circuit.high
+ gate_minus * circuit.low
- (circuit.low + circuit.high) * circuit.threshold_voltage
)
)
denominator = (
circuit.fixed_conductance
+ scale * (
gate_plus + gate_minus - 2.0 * circuit.threshold_voltage
)
)
if denominator <= 0.0:
raise ValueError("nonpositive effective conductance")
return float(numerator / denominator)
def solution_line(circuit: Circuit, task: Task) -> tuple[float, float]:
"""Return slope/intercept of gate_plus versus gate_minus at zero error."""
label = task.label_voltage
scale = circuit.conductance_per_gate
denominator = scale * (circuit.high - label)
if denominator == 0.0:
raise ValueError("label coincides with high boundary")
slope = -(circuit.low - label) / (circuit.high - label)
intercept = -(
circuit.fixed_conductance * (task.input_voltage - label)
+ scale * circuit.threshold_voltage
* (2.0 * label - circuit.low - circuit.high)
) / denominator
return float(slope), float(intercept)
def joint_solution(circuit: Circuit, tasks: Iterable[Task]) -> Array:
tasks = tuple(tasks)
if len(tasks) != 2:
raise ValueError("joint_solution expects exactly two tasks")
slope_a, intercept_a = solution_line(circuit, tasks[0])
slope_b, intercept_b = solution_line(circuit, tasks[1])
if slope_a == slope_b:
raise ValueError("parallel solution lines have no unique joint solution")
gate_minus = (intercept_b - intercept_a) / (slope_a - slope_b)
return np.asarray(
(gate_minus, slope_a * gate_minus + intercept_a), dtype=float)
def voltage_drop_squares(circuit: Circuit, output: float) -> Array:
return np.asarray(
((output - circuit.low) ** 2, (circuit.high - output) ** 2),
dtype=float,
)
def standard_clean_rate(
circuit: Circuit, gates: Array, task: Task, nudging: float = 1.0
) -> tuple[Array, float, float]:
output_free = free_output(circuit, gates, task.input_voltage)
output_clamped = output_free + nudging * (
task.label_voltage - output_free)
rate = circuit.measured_learning_rate * (
voltage_drop_squares(circuit, output_free)
- voltage_drop_squares(circuit, output_clamped)
)
return rate, output_free, output_clamped
def overclamped_clean_rate(
circuit: Circuit,
gates: Array,
task: Task,
*,
nudging: float = 0.25,
clamp_magnitude: Optional[float] = None,
) -> tuple[Array, float, float]:
"""Leading-order overclamping signal from Appendix F, Eq. F6--F8."""
output_free = free_output(circuit, gates, task.input_voltage)
error = task.label_voltage - output_free
magnitude = circuit.high if clamp_magnitude is None else clamp_magnitude
output_clamped = output_free + nudging * magnitude * np.sign(error)
rate = circuit.measured_learning_rate * (
voltage_drop_squares(circuit, output_free)
- voltage_drop_squares(circuit, output_clamped)
)
return rate, output_free, output_clamped
def calibrate_predictor(
predictor: LocalPredictor,
states: Array,
measurement: Callable[[Array], Array],
*,
epochs: int,
learning_rate: float,
) -> int:
"""Sequential local LMS calibration; returns neutral observation count."""
states = np.asarray(states, dtype=float)
if states.ndim != 2 or states.shape[1] != 2:
raise ValueError("calibration states must have shape (observations, 2)")
if epochs < 1:
raise ValueError("epochs must be positive")
count = 0
for _ in range(epochs):
for gates in states:
predictor.update(gates, measurement(gates), learning_rate)
count += 1
return count
def local_replay_update(
predictor: LocalPredictor,
gates: Array,
teaching_measurement: Array,
eligibility: Array,
learning_rate: float,
) -> Array:
"""The complete stored-tuple SDIL update, independent of any task/model."""
residual = np.asarray(teaching_measurement) - predictor.predict(gates)
return learning_rate * residual * np.asarray(eligibility)
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