Spaces:
Running on Zero
Running on Zero
File size: 7,710 Bytes
9936912 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | #!/usr/bin/env python3
"""Base STEM and Mathematical Reasoning replay dataset generator.
Prevents catastrophic forgetting by mixing in step-by-step Chain of Thought (CoT)
linear algebra, calculus, ODE, Laplace transforms, and physical systems reasoning.
"""
from __future__ import annotations
import math
import sys
from pathlib import Path
import numpy as np
PROJECT_ROOT = Path(__file__).resolve().parents[1]
if str(PROJECT_ROOT) not in sys.path:
sys.path.insert(0, str(PROJECT_ROOT))
from controlai_data.schema import make_record
def n(val: float) -> str:
if abs(val) < 1e-12:
return "0.0"
return f"{val:.6g}"
def mat_str(arr: np.ndarray) -> str:
return repr(np.asarray(arr, dtype=float).tolist())
def generate_laplace_partial_fractions(count: int, seed: int) -> list[dict]:
"""Generate partial fraction expansion of rational transfer functions with step-by-step math."""
records = []
for i in range(count):
p1 = -(1.0 + 0.5 * (i % 4))
p2 = -(3.0 + 0.5 * (i % 5))
z = 2.0 + 0.25 * (i % 3)
# F(s) = (s + z) / ((s - p1)(s - p2)) = A / (s - p1) + B / (s - p2)
# A = (p1 + z) / (p1 - p2)
# B = (p2 + z) / (p2 - p1)
A_res = (p1 + z) / (p1 - p2)
B_res = (p2 + z) / (p2 - p1)
prompt = (
f"Compute the inverse Laplace transform $f(t) = \\mathcal{{L}}^{{-1}}\\{{F(s)\\}}$ for $t \\ge 0$, where "
f"$$F(s) = \\frac{{s + {n(z)}}}{{(s + {n(-p1)})(s + {n(-p2)})}}$$. "
f"Show the step-by-step partial fraction expansion and state the time-domain signal."
)
answer = (
f"### 1. Partial Fraction Decomposition\n"
f"Write $F(s)$ in terms of first-order partial fractions with distinct poles $s_1 = {n(p1)}$ and $s_2 = {n(p2)}$:\n"
f"$$F(s) = \\frac{{s + {n(z)}}}{{(s - ({n(p1)}))(s - ({n(p2)}))}} = \\frac{{C_1}}{{s - ({n(p1)})}} + \\frac{{C_2}}{{s - ({n(p2)})}}$$\n\n"
f"### 2. Residue Calculation\n"
f"Applying the residue formula:\n"
f"$$C_1 = \\lim_{{s \\to {n(p1)}}} (s - ({n(p1)})) F(s) = \\frac{{{n(p1)} + {n(z)}}}{{{n(p1)} - ({n(p2)})}} = \\frac{{{n(p1 + z)}}}{{{n(p1 - p2)}}} = {n(A_res)}$$\n"
f"$$C_2 = \\lim_{{s \\to {n(p2)}}} (s - ({n(p2)})) F(s) = \\frac{{{n(p2)} + {n(z)}}}{{{n(p2)} - ({n(p1)})}} = \\frac{{{n(p2 + z)}}}{{{n(p2 - p1)}}} = {n(B_res)}$$\n\n"
f"Thus, the expanded expression is:\n"
f"$$F(s) = \\frac{{{n(A_res)}}}{{s + {n(-p1)}}} + \\frac{{{n(B_res)}}}{{s + {n(-p2)}}}$$\n\n"
f"### 3. Inverse Laplace Transform\n"
f"Using the standard pair $\\mathcal{{L}}^{{-1}}\\{{\\frac{{1}}{{s + a}}\\}} = e^{{-at}} u(t)$:\n"
f"$$f(t) = ({n(A_res)} e^{{{n(p1)} t}} + {n(B_res)} e^{{{n(p2)} t}}) u(t)$$\n"
f"**Conclusion:** The causal time-domain signal is $f(t) = {n(A_res)} e^{{{n(p1)} t}} + {n(B_res)} e^{{{n(p2)} t}}$ for $t \\ge 0$."
)
record = make_record(
record_id=f"stem_laplace_partial_fraction_{i:04d}",
domain="classical_optimal_control",
family="stem_laplace_partial_fraction",
task_type="derivation",
difficulty="foundation",
template_id=f"laplace_partial_fraction_prompt_{i % 4}",
prompt=prompt,
answer=answer,
ground_truth={
"kind": "laplace_inversion",
"p1": p1,
"p2": p2,
"z": z,
"C1": A_res,
"C2": B_res,
},
source_refs=["stanford_ee263_course_reader", "astrom_murray_feedback_systems_1e"],
verifier="verify_laplace_inversion",
tool="requirements_analysis",
)
records.append(record)
return records
def generate_rlc_circuit_cases(count: int, seed: int) -> list[dict]:
"""Generate series RLC circuit frequency response & damping ratio analysis."""
records = []
for i in range(count):
R = 10.0 + 2.0 * (i % 5) # Ohms
L = 0.05 + 0.01 * (i % 4) # Henry
C = 0.001 + 0.0002 * (i % 3) # Farad
# Series RLC: L * ddot(q) + R * dot(q) + (1/C)*q = v(t)
# Characteristic eq: s^2 + (R/L)s + 1/(LC) = s^2 + 2*zeta*wn*s + wn^2
wn = 1.0 / math.sqrt(L * C)
zeta = R / (2.0 * math.sqrt(L / C))
wd = wn * math.sqrt(abs(1.0 - zeta**2)) if zeta < 1.0 else 0.0
if zeta < 1.0:
regime = "underdamped"
elif abs(zeta - 1.0) < 1e-4:
regime = "critically damped"
else:
regime = "overdamped"
prompt = (
f"Analyze a series RLC circuit with resistor $R = {n(R)}$ $\\Omega$, inductor $L = {n(L)}$ H, "
f"and capacitor $C = {n(C)}$ F. "
f"Find the natural frequency $\\omega_n$, damping ratio $\\zeta$, damped natural frequency $\\omega_d$, "
f"and characterize the transient response regime."
)
answer = (
f"### 1. Differential Equation and Standard Form\n"
f"Applying Kirchhoff's Voltage Law (KVL) around the series loop with charge $q(t)$:\n"
f"$$L \\frac{{d^2q}}{{dt^2}} + R \\frac{{dq}}{{dt}} + \\frac{{1}}{{C}} q = v(t)$$\n\n"
f"Dividing by $L$ puts the equation into canonical second-order form $\\ddot{{q}} + 2\\zeta \\omega_n \\dot{{q}} + \\omega_n^2 q = \\frac{{1}}{{L}} v(t)$:\n"
f"$$\\ddot{{q}} + \\frac{{{n(R)}}}{{{n(L)}}} \\dot{{q}} + \\frac{{1}}{{{n(L)} \\times {n(C)}}} q = \\frac{{1}}{{{n(L)}}} v(t)$$\n\n"
f"### 2. Parameter Extraction\n"
f"1. **Natural frequency $\\omega_n$**:\n"
f"$$\\omega_n = \\frac{{1}}{{\\sqrt{{LC}}}} = \\frac{{1}}{{\\sqrt{{{n(L)} \\times {n(C)}}}}} = {n(wn)}\\text{{ rad/s}}$$\n\n"
f"2. **Damping ratio $\\zeta$**:\n"
f"$$2\\zeta \\omega_n = \\frac{{R}}{{L}} \\implies \\zeta = \\frac{{R}}{{2 \\sqrt{{L/C}}}} = \\frac{{{n(R)}}}{{2 \\sqrt{{{n(L)} / {n(C)}}}}} = {n(zeta)}$$\n\n"
f"3. **Damped frequency $\\omega_d$**:\n"
f"$$\\omega_d = \\omega_n \\sqrt{{|1 - \\zeta^2|}} = {n(wd)}\\text{{ rad/s}}$$\n\n"
f"### 3. Conclusion\n"
f"Since $\\zeta = {n(zeta)}$ ({'< 1' if zeta < 1.0 else '> 1'}), the system is **{regime}**."
)
record = make_record(
record_id=f"stem_rlc_circuit_{i:04d}",
domain="classical_optimal_control",
family="stem_rlc_circuit_transient",
task_type="numerical",
difficulty="foundation",
template_id=f"rlc_circuit_prompt_{i % 4}",
prompt=prompt,
answer=answer,
ground_truth={
"kind": "second_order_transient",
"R": R,
"L": L,
"C": C,
"wn": wn,
"zeta": zeta,
"wd": wd,
"regime": regime,
},
source_refs=["stanford_ee263_course_reader", "astrom_murray_feedback_systems_1e"],
verifier="verify_second_order_transient",
tool="requirements_analysis",
)
records.append(record)
return records
def build_all_stem_records(count_per_case: int = 50, seed: int = 20260902) -> list[dict]:
"""Compile all STEM replay records."""
records = []
records.extend(generate_laplace_partial_fractions(count_per_case, seed))
records.extend(generate_rlc_circuit_cases(count_per_case, seed + 1000))
return records
if __name__ == "__main__":
cases = build_all_stem_records(40)
print(f"Generated {len(cases)} STEM replay SFT records.")
|