L1 ⚛ L1-195 ⊙ Testnet

Transient Heat Conduction — inverse heat conduction (IHCP)

Heat Transfer · Parabolic PDE inverse source/BC estimation

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⬇ L1-195.md

# ⚛ L1 Principle — Transient Heat Conduction — inverse heat conduction (IHCP) **ID:** `L1-195` · **Status:** ⊙ Testnet (genesis catalog) > **🌐 Domain:** Heat Transfer — *Parabolic PDE inverse source/BC estimation* > **🎯 Problem class:** linear inverse · **🧮 Solution space:** 3D temperature field spacetime > **📡 Carrier:** thermal · **🌫 Noise:** gaussian > **⚖ Difficulty (δ):** 5 · **⛓ Block:** 41554033 --- ## 🧠 1. Introduction **Transient Heat Conduction — inverse heat conduction (IHCP)** is a **linear inverse problem** whose unknown lives in **3D temperature field spacetime** space, within the **Parabolic PDE inverse source/BC estimation** sub-domain of **Heat Transfer**. Measurements consist of thermal radiation / temperature measurements via a **thermocouples** sensing mechanism. The forward operator applies, in order: L · initial condition operator; L · parabolic propagate operator; S · record · sensors operator; detector accumulates flux over the exposure window. Observations are corrupted by additive Gaussian noise. IHCP is ill-posed (Hadamard): high-frequency BC content decays exponentially. Requires regularization (Tikhonov, future-time-stepping, PINN). ## ⚙ 2. Forward Model Physical chain: **x** → L · initial condition → L · parabolic propagate → S · record · sensors → Temporal integration → **y** (detector). ``` y = ∫_t dt `S.record.sensors` `L.parabolic_propagate` `L.initial_condition` x + n, n ~ 𝒩(0, σ²) ``` **Measurement DAG:** | Primitive | What it does | |---|---| | `L.initial_condition` | L · initial condition operator | | `L.parabolic_propagate` | L · parabolic propagate operator | | `S.record.sensors` | S · record · sensors operator | | `int.temporal` | Detector accumulates flux over the exposure window | ## 🔬 3. Physics Fingerprint | Property | Value | |---|---| | Domain | Heat Transfer | | Sub domain | Parabolic PDE inverse source/BC estimation | | Carrier | thermal | | Problem class | linear_inverse | | Solution space | 3D_temperature_field_spacetime | | Noise model | gaussian | | Integration axis | temporal | | Difficulty delta | 5 | | L dag | 3 | ## 📡 4. Measurement Model IHCP is ill-posed (Hadamard): high-frequency BC content decays exponentially. Requires regularization (Tikhonov, future-time-stepping, PINN). | Metric | Value | |---|---| | Metric | PSNR_dB | | Secondary | SSIM | ## 📏 5. Operating Range (Ω) **Center problem class:** `ihcp` · **Forward operator:** `ihcp_forward` **Center point:** | Parameter | Unit | Value | |---|---|---| | H | px | 64 | | W | px | 64 | | Z | — | 64 | | Fs hz | Hz | 10 | | Snr db | dB | 30 | | K error | — | 0 | | Bc error | — | 0 | | N samples | — | 200 | | N sensors | — | 16 | | Rho cp error | — | 0 | | Sensor pos error | — | 0 | **Allowed bounds:** | Parameter | Unit | Range | |---|---|---| | H | px | 16 – 512 | | W | px | 16 – 512 | | Z | — | 8 – 256 | | Fs hz | Hz | 0.1 – 1000 | | Snr db | dB | 0 – 40 | | K error | — | 0.0 – 0.1 | | Bc error | — | 0.0 – 0.5 | | N samples | — | 20 – 10000 | | N sensors | — | 4 – 256 | | Rho cp error | — | 0.0 – 0.1 | | Sensor pos error | — | 0.0 – 0.02 | ## 🎯 6. Tolerance (ε) **Center tolerance:** 25.0 | Metric | Range | |---|---| | Psnr db | 10.0 – 45.0 | ## ⚖ 7. Hardness Function Hardness scales as **`epsilon_fn`** on **PSNR_dB**, with κ = `800` and δ = `5`. ## 💾 8. Reference Dataset - **primary** · weight 1.0 · IPFS _(not pinned yet)_ ## 9. On-chain Registration - **Chain hash:** `0x366622440a8eb6dc6b7aadc3798ac1d2dc997b5aa608e00baf504c2d99c265eb` - **Chain tx hash:** `0x0a179d9695a8fc81fd7610fafc8d16e723fac4c7e41bcf3ca5cac6074d737ae0` - **Chain block:** `41554033` --- ## File Mapping This bundle consists of: `L1-195.md`, `L1-195.json`. | File | Role | How to regenerate | |------|------|-------------------| | `L1-195.md` | Source of truth — edit this | Human or LLM | | `L1-195.json` | Structured metadata for the registry | LLM regenerates from the sections above | **Prompt for your LLM after editing this Markdown:** > Read the attached Markdown. Regenerate the sibling `.json` so every field matches. > Preserve the schema documented in the rows above. > Output each file in its own fenced code block tagged with the filename. > Output only the JSON object. _This Markdown was auto-synthesized from the catalog row for `L1-195`._ _Edit it, regenerate the JSON, and submit at [/submit](/submit) to claim the artifact._

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Regenerate L1-195.json so every field matches the Markdown.
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L2-195-001 Transient Heat Conduction — inverse heat conduction (IHCP) Mismatch-Only Spec

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📋 JSON metadata

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  "chain_block": 41554033,
  "chain_hash": "0x366622440a8eb6dc6b7aadc3798ac1d2dc997b5aa608e00baf504c2d99c265eb",
  "chain_tx_hash": "0x0a179d9695a8fc81fd7610fafc8d16e723fac4c7e41bcf3ca5cac6074d737ae0",
  "domain": "Heat Transfer",
  "hardness_fn": {
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    "metric": "PSNR_dB",
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    "regime": "IHCP is ill-posed (Hadamard): high-frequency BC content decays exponentially. Requires regularization (Tikhonov, future-time-stepping, PINN).",
    "secondary": "SSIM"
  },
  "physics_fingerprint": {
    "L_DAG": 3.0,
    "carrier": "thermal",
    "difficulty_delta": 5,
    "domain": "Heat Transfer",
    "integration_axis": "temporal",
    "noise_model": "gaussian",
    "primitives": [
      "L.initial_condition",
      "L.parabolic_propagate",
      "S.record.sensors",
      "int.temporal"
    ],
    "problem_class": "linear_inverse",
    "sensing_mechanism": "thermocouples",
    "solution_space": "3D_temperature_field_spacetime",
    "sub_domain": "Parabolic PDE inverse source/BC estimation",
    "title": "Transient Heat Conduction \u2014 inverse heat conduction (IHCP)"
  },
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    "center_spec": {
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      "input_format": "measurement_only",
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        "N_sensors": 16,
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        "W": 64,
        "Z": 64,
        "bc_error": 0.0,
        "fs_Hz": 10,
        "k_error": 0.0,
        "rho_cp_error": 0.0,
        "sensor_pos_error": 0.0
      },
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      "N_sensors": [
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      ],
      "SNR_dB": [
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      ],
      "W": [
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      ],
      "Z": [
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      ],
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      "k_error": [
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      "rho_cp_error": [
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      ],
      "sensor_pos_error": [
        0.0,
        0.02
      ]
    }
  },
  "staked_pwm": 0.0,
  "status": "testnet",
  "sub_domain": "Parabolic PDE inverse source/BC estimation",
  "title": "Transient Heat Conduction \u2014 inverse heat conduction (IHCP)"
}

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