Experiment Report: burgers (pysr)

Metadata

Algorithm	pysr
Dataset	burgers
Expression	\( u_{t} = - 1.031 u u_{x} e^{u u_{xx}} \)
Best NMSE	0.132239
R²	0.867761
NMSE	0.1322
Iterations	1
Early Stopped	False
config.algorithm	pysr
config.library_fingerprint	3d7bbc2cc5b3e98b
config.terms	['u', 'u_x', 'u_xx']
config.seed	0
config.niterations	40
config.population_size	33
config.populations	15
config.maxsize	20
config.binary_operators	['+', '-', '*', '/']
config.unary_operators	['sin', 'cos', 'exp', 'log']
config.extra_pysr_kwargs	{'deterministic': True, 'parallelism': 'serial'}
config.provider_kind	finite_diff

Convergence

Read the best native score reached at each search iteration.

Parity

Compare predicted and observed targets against the diagonal of exact agreement.

Equation

Read the published equation with its fitted coefficients and active terms.

Equation Tree

Read the operators and derivatives that compose the published equation.

Pareto Front Table

Compare candidate complexity, loss and fitted outer scale alongside each expression.

Residual

Read the distribution and spatial pattern of predicted minus observed target values.

Coefficient Bar

Compare the magnitudes and signs of the active fitted coefficients.

Field Comparison

Compare the measured field with forward integration of the discovered equation and their difference.

Pde Residual Field

Compare the fitted derivative target with the equation prediction and their residual, without forward integration.

Time Slices

Compare measured and integrated fields at the labeled observation times.

Error Heatmap

Locate the signed field error from forward integration at the labeled physical coordinates.

Plugin Plots

Pareto Front

PySR's own complexity/loss Pareto front; the selected model is highlighted.

KD Audit Path

KD's independent NMSE re-score per Pareto complexity (the audit substrate); the selected model is highlighted.

Score Agreement

PySR loss vs KD NMSE per Pareto point -- do the two rankings agree?

Warnings

Protected-operator note: the integrated RHS contains exp, which the platform evaluates with protected semantics (safe_*/clamped wrappers) — the same semantics under which the equation was scored during search. Trajectories that would diverge under bare operators may remain bounded.

solve_ivp integration failed: Required step size is less than spacing between numbers.

Integration-dependent plots (field comparison, time slices, error heatmap) are rendered in degraded form; other plots and metrics are unaffected.

Field comparison, time slices and error heatmap were regenerated with corrected periodic spatial boundaries. The equation, coefficients and search-fit metrics remain those of the original run; no search was repeated.

Full result (JSON, for debugging) — click to expand

{
  "result_schema_version": 1,
  "best_expression": "mul(u, mul(u_x, exp(mul(u, u_xx))))",
  "best_score": 0.13223945797483555,
  "iterations": 1,
  "early_stopped": false,
  "final_eval": {
    "mse": 0.0009028244062865235,
    "nmse": 0.13223945797483555,
    "r2": 0.8677605420251633,
    "score": -360703.6584597962,
    "complexity": 1,
    "coefficients": [
      -1.0314385890960693
    ],
    "is_valid": true,
    "error_message": "",
    "invalid_reason": null,
    "selected_indices": null,
    "residuals": "[51456 elements]",
    "terms": [
      "mul(u, mul(u_x, exp(mul(u, u_xx))))"
    ],
    "expression": "mul(u, mul(u_x, exp(mul(u, u_xx))))",
    "lhs_name": null,
    "condition_number": null,
    "condition_number_computed": false
  },
  "actual": "[51456 elements]",
  "predicted": "[51456 elements]",
  "dataset_name": "burgers",
  "algorithm_name": "PySRPlugin",
  "config": {
    "algorithm": "pysr",
    "library_fingerprint": "3d7bbc2cc5b3e98b",
    "terms": [
      "u",
      "u_x",
      "u_xx"
    ],
    "seed": 0,
    "niterations": 40,
    "population_size": 33,
    "populations": 15,
    "maxsize": 20,
    "binary_operators": [
      "+",
      "-",
      "*",
      "/"
    ],
    "unary_operators": [
      "sin",
      "cos",
      "exp",
      "log"
    ],
    "extra_pysr_kwargs": {
      "deterministic": true,
      "parallelism": "serial"
    },
    "provider_kind": "finite_diff"
  },
  "recorder": {
    "pareto_complexity": [
      [
        3,
        4,
        5,
        6,
        7,
        8,
        9,
        10,
        11,
        12,
        13,
        14,
        15,
        16
      ]
    ],
    "pareto_loss": [
      [
        0.005643713,
        0.0055964896,
        0.0051963115,
        0.0035915908,
        0.0033166532,
        0.0019214299,
        0.0017690312,
        0.0009028242,
        0.00074612413,
        0.00068505656,
        0.0006794184,
        0.0006649254,
        0.0006597807,
        0.00065867044
      ]
    ],
    "pareto_nmse": [
      [
        0.8266520925942668,
        0.8197350390249417,
        0.7611197213334521,
        0.5260713925301798,
        0.48580042381292476,
        0.2814377824239626,
        0.25911546582053996,
        0.13223945797483555,
        0.10928711675971098,
        0.1003423325645523,
        0.09944705677516366,
        0.09737270192855248,
        0.09652516603386391,
        0.0964771841128685
      ]
    ],
    "pareto_expressions": [
      [
        "u",
        "sin(u)",
        "mul(u, u_x)",
        "mul(u, sin(u_x))",
        "mul(u, sin(sin(u_x)))",
        "mul(u, mul(u_x, exp(u_x)))",
        "mul(u, mul(exp(u_x), sin(u_x)))",
        "mul(u, mul(u_x, exp(mul(u, u_xx))))",
        "mul(u, mul(exp(mul(u, u_xx)), sin(u_x)))",
        "mul(u, mul(u_x, exp(mul(-1, mul(0.20801155, n2(u_xx))))))",
        "mul(u, sin(mul(0.9717037912578143, mul(u_x, exp(mul(-1, mul(0.20449287, n2(u_xx))))))))",
        "mul(u, mul(u_x, mul(exp(mul(-1, mul(0.16872406, n2(u_xx)))), exp(mul(0.16872406, mul(u, u_xx))))))",
        "mul(u, sin(mul(0.9904688, mul(u_x, mul(exp(mul(-1, mul(0.16872406, n2(u_xx)))), exp(mul(0.16872406, mul(u, u_xx))))))))",
        "mul(u, mul(u_x, mul(exp(mul(-1, mul(0.1475392, n2(u_xx)))), exp(mul(0.2524565192877724, mul(u, u_xx))))))"
      ]
    ],
    "selected_complexity": [
      10
    ],
    "selected_loss": [
      0.0009028242
    ],
    "selected_nmse": [
      0.13223945797483555
    ],
    "_best_score": [
      0.13223945797483555
    ],
    "_best_expr": [
      "mul(u, mul(u_x, exp(mul(u, u_xx))))"
    ],
    "_n_candidates": [
      14
    ]
  },
  "lhs_label": "u_t",
  "equation": {
    "form": "EVOLUTION",
    "lhs_spec": {
      "field": "u",
      "axis": "t",
      "order": 1
    },
    "terms": [
      [
        "mul(u, mul(u_x, exp(mul(u, u_xx))))",
        {
          "kind": "Scalar",
          "value": -1.0314386439784
        }
      ]
    ],
    "attrs": null,
    "active_indices": null
  },
  "manifest": {
    "dataset_cache_fingerprint": "burgers:grid:u:t:lhs_order=1:x,t:periodic=t=N,x=N_u(256, 201)_fb43be53",
    "kd_version": "0.7.5",
    "seed": 0,
    "terms": [
      "u",
      "u_x",
      "u_xx"
    ],
    "artifacts": null,
    "resumed": false,
    "resume_source": null
  },
  "run_record": {
    "schema_version": 2,
    "evidence_hash_scheme": "kd-evidence-v1",
    "created_at": "2026-09-14T05:03:25+00:00",
    "cost": {
      "wallclock_seconds": 1597.751478340244,
      "search_seconds": 1597.7377895100508,
      "boundary_results": 14,
      "boundary_invalid_results": 0,
      "preprocessing_seconds": 0.013688830193132162,
      "surrogate_train_seconds": null,
      "cpu_seconds": 3119.7868390000003,
      "tokens_in": null,
      "tokens_out": null,
      "tokens_cached": null,
      "api_cost_usd": null
    },
    "evidence": {
      "instrument": "pysr",
      "dataset_name": "burgers",
      "dataset_cache_fingerprint": "burgers:grid:u:t:lhs_order=1:x,t:periodic=t=N,x=N_u(256, 201)_fb43be53",
      "seed": 0,
      "is_valid": true,
      "expression": "mul(u, mul(u_x, exp(mul(u, u_xx))))",
      "score_kind": "NMSE",
      "score_direction": "min",
      "headline_coefficient_source": "platform_refit",
      "catalog_fit": {
        "form": "EVOLUTION",
        "lhs_spec": {
          "field": "u",
          "axis": "t",
          "order": 1
        },
        "terms": [
          [
            "mul(u, mul(u_x, exp(mul(u, u_xx))))",
            {
              "kind": "Scalar",
              "value": -1.0314386439784
            }
          ]
        ],
        "attrs": null,
        "active_indices": null
      },
      "support": [
        "mul(u, mul(u_x, exp(mul(u, u_xx))))"
      ],
      "coefficients": [
        -1.0314386439784
      ],
      "complexity": 1,
      "mse": 0.0009028244062865235,
      "nmse": 0.13223945797483555,
      "r2": 0.8677605420251633,
      "score": 0.13223945797483555,
      "residual_stats": {
        "mean": -1.2885493684720935e-09,
        "std": 0.03004732853664172,
        "max_abs": 0.27794765363898594,
        "n": 51456
      },
      "invalid_reason": null,
      "error_detail": null
    },
    "evidence_hash": "a98ae1752bd64a616240d06672e5f9bdd000f56183458c381d12654830fda6fa",
    "run_spec": {
      "kd_version": "0.7.5",
      "config": {
        "algorithm": "pysr",
        "binary_operators": [
          "+",
          "-",
          "*",
          "/"
        ],
        "extra_pysr_kwargs": {
          "deterministic": true,
          "parallelism": "serial"
        },
        "library_fingerprint": "3d7bbc2cc5b3e98b",
        "maxsize": 20,
        "niterations": 40,
        "population_size": 33,
        "populations": 15,
        "provider_kind": "finite_diff",
        "seed": 0,
        "terms": [
          "u",
          "u_x",
          "u_xx"
        ],
        "unary_operators": [
          "sin",
          "cos",
          "exp",
          "log"
        ]
      },
      "dataset_cache_fingerprint": "burgers:grid:u:t:lhs_order=1:x,t:periodic=t=N,x=N_u(256, 201)_fb43be53",
      "library_fingerprint": "3d7bbc2cc5b3e98b",
      "artifacts": null,
      "config_canon_scheme": "kd-config-v1"
    },
    "run_spec_hash": "5b1f9214e9a4a52534ffe5ef36e6a0486b8a04c2702710f64a5e8c237b98bbe0",
    "run_spec_hash_scheme": "kd-runspec-v1",
    "record_hash": "435b22a70ef37bb350a5875fa07a65442fd19da8da870f5b73436e5c33db976c",
    "record_hash_scheme": "kd-record-v1"
  },
  "score_kind": "NMSE",
  "score_direction": "min"
}