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"# Scientific ML and Operator Learning\n",
"\n",
"*The Mathematics of Large Language Models ยท Chapter 13*\n",
"\n",
"Test laws, conditions and trajectory evidence separately.\n",
"\n",
"These pages illustrate the mathematics. Read the saved figures and calculations in order, or open [the interactive browser edition](reader.html) to change the examples. No code editing is needed.\n",
"\n",
"Request the quantity and units, governing relation, domain, material parameters, initial and boundary conditions, observation locations and reference evidence. Return a model-assumption sheet, separate PDE/boundary/initial residuals and a comparison with an analytic case when one applies. For function inputs specify the basis, coefficients and time, and distinguish observation-grid density from model validity. For law discovery require trajectory and derivative provenance, candidate library, column scaling and the precise sparse objective; return fitted coefficients, derivative residuals and an independent trajectory comparison."
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"Reproducing these calculations
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"\n",
"The optional calculation cells use Python with NumPy, SciPy, and Matplotlib. All inputs are included in this file. The figures below are saved, so running these cells is optional.\n",
"\n",
"Show the answer
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"\n",
"Four times faster: its curvature is -(2 pi)^2 u = -4 pi^2 u, so it decays like exp(-4 pi^2 t).\n",
"\n",
"Show the answer
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"\n",
"A constant shift changes neither u_t nor u_xx, so the PDE passes. Both ends and the start profile are off by 0.2, giving mean squared residual 0.2^2 = 0.04 each.\n",
"\n",
"Show the answer
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"\n",
"Mode 2 is at 1% when 4 pi^2 t = ln 100 = 4.61, so t = 0.117. Then mode 1 keeps exp(-pi^2 x 0.117) = 0.32.\n",
"\n",
"Show the answer
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"\n",
"The size of the coefficients and a rollout against held-out data: right terms with shrunken coefficients still predict the wrong motion.\n",
"\n",
"Show the answer
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"\n",
"More accurate: at k = 5 the new kernel gives 1/26 = 0.038 versus 0.265 here, so every dropped mode carries less energy.\n",
"\n",
"Show the answer
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"\n",
"About 3,950 times: 1 + (20 pi)^2 = 1 + 3948 = 3949.\n",
"\n",
"Show the answer
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"\n",
"0.25: the operator is linear in f, so the peak doubles from 1/8 to 1/4.\n",
"\n",
"Show the answer
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"\n",
"With one shape every output is a multiple of that shape, so all outputs peak in the same place; moving the peak needs more shapes.\n",
"\n",
"