{
  "article_slug": "physics-informed-residual-scales",
  "figures": [
    {
      "id": "penalty-unit-scale",
      "title": "Residual scaling changes the optimum",
      "kind": "analytical_curve",
      "x_label": "Residual scale s (dimensionless)",
      "y_label": "Parameter / unscaled absolute error",
      "x": {
        "name": "Residual scale s (dimensionless)",
        "values": [
          0.1,
          0.2,
          0.5,
          1,
          2,
          5,
          10
        ]
      },
      "series": [
        {
          "name": "Optimal theta",
          "values": [
            0.9900990099009901,
            0.9615384615384615,
            0.8,
            0.5,
            0.2,
            0.038461538461538464,
            0.009900990099009901
          ]
        },
        {
          "name": "Data mismatch",
          "values": [
            0.00990099009900991,
            0.03846153846153855,
            0.19999999999999996,
            0.5,
            0.8,
            0.9615384615384616,
            0.9900990099009901
          ]
        },
        {
          "name": "Compensated theta",
          "values": [
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5,
            0.5
          ]
        }
      ],
      "caption": "Constructed inconsistent scalar objective (theta-1)^2+(s theta)^2. Compensating the residual weight by 1/s^2 preserves theta=0.5. No neural benchmark."
    },
    {
      "id": "poisson-mode-error",
      "title": "Equal residual norm, unequal field error",
      "kind": "analytical_curve",
      "x_label": "Positive Fourier-sine mode k",
      "y_label": "L2 norm (dimensionless)",
      "x": {
        "name": "Positive Fourier-sine mode k",
        "values": [
          1,
          2,
          3,
          4,
          6,
          8,
          12
        ]
      },
      "series": [
        {
          "name": "Field error, residual norm=1",
          "values": [
            0.10132118364233778,
            0.025330295910584444,
            0.011257909293593086,
            0.006332573977646111,
            0.0028144773233982714,
            0.0015831434944115277,
            0.0007036193308495678
          ]
        },
        {
          "name": "Gradient error, residual norm=1",
          "values": [
            0.3183098861837907,
            0.15915494309189535,
            0.1061032953945969,
            0.07957747154594767,
            0.05305164769729845,
            0.039788735772973836,
            0.026525823848649224
          ]
        }
      ],
      "caption": "Constructed homogeneous Dirichlet Poisson modes on [0,1]. Residual L2 norm fixed to one; field and gradient error scale as (k pi)^-2 and (k pi)^-1. No physical measurements."
    }
  ]
}