| Algorithm | pysindy |
|---|---|
| Dataset | burgers |
| Expression | \( u_{t} = - u u_{x} + 0.1002 u_{xx} \) |
| Best NMSE | 1.11783e-05 |
| R² | 0.999989 |
| NMSE | 1.118e-05 |
| Iterations | 1 |
| Early Stopped | False |
| config.algorithm | pysindy |
| config.library_fingerprint | 017dbdceae9f468d |
| config.terms | ['u', 'u_x', 'u_xx', 'u*u_x'] |
| config.threshold | 0.05 |
| config.max_iter | 20 |
| config.normalize_columns | False |
| config.unbias | True |
| config.seed | 0 |
| config.extra_optimizer_kwargs | None |
| config.provider_kind | finite_diff |
Read the best native score reached at each search iteration.
Compare predicted and observed targets against the diagonal of exact agreement.
Read the published equation with its fitted coefficients and active terms.
Read the operators and derivatives that compose the published equation.
Read the distribution and spatial pattern of predicted minus observed target values.
Compare the magnitudes and signs of the active fitted coefficients.
Compare the measured field with forward integration of the discovered equation and their difference.
Compare the fitted derivative target with the equation prediction and their residual, without forward integration.
Compare measured and integrated fields at the labeled observation times.
Locate the signed field error from forward integration at the labeled physical coordinates.
pysindy's native-coefficient NMSE vs kd's independent refit re-score for the one selected STLSQ model -- do the two scores agree? Support size annotated.
{
"result_schema_version": 1,
"best_expression": "add(u_xx, mul(u,u_x))",
"best_score": 1.1178326915795506e-05,
"iterations": 1,
"early_stopped": false,
"final_eval": {
"mse": 7.631660410276492e-08,
"nmse": 1.1178326915795506e-05,
"r2": 0.9999888216730842,
"score": -843276.2397612393,
"complexity": 2,
"coefficients": [
0.0,
0.0,
0.10018764436244965,
-1.0000022649765015
],
"is_valid": true,
"error_message": "",
"invalid_reason": null,
"selected_indices": [
2,
3
],
"residuals": "[51456 elements]",
"terms": [
"u",
"u_x",
"u_xx",
"mul(u,u_x)"
],
"expression": "add(u_xx, mul(u,u_x))",
"lhs_name": null,
"condition_number": null,
"condition_number_computed": false
},
"actual": "[51456 elements]",
"predicted": "[51456 elements]",
"dataset_name": "burgers",
"algorithm_name": "PySINDyPlugin",
"config": {
"algorithm": "pysindy",
"library_fingerprint": "017dbdceae9f468d",
"terms": [
"u",
"u_x",
"u_xx",
"mul(u,u_x)"
],
"threshold": 0.05,
"max_iter": 20,
"normalize_columns": false,
"unbias": true,
"seed": 0,
"extra_optimizer_kwargs": null,
"provider_kind": "finite_diff"
},
"recorder": {
"native_nmse": [
1.1178326915795506e-05
],
"refit_nmse": [
1.1178326915795547e-05
],
"support_size": [
2
],
"stlsq_history": [
[
[
8.639945813858149e-05,
2.1504699826424556e-09,
0.10021960833583707,
-1.0002458992828476
],
[
0.0,
0.0,
0.10020081617748726,
-1.0000694092622109
],
[
0.0,
0.0,
0.10017314688254167,
-0.9998584575380975
]
]
],
"_best_score": [
1.1178326915795506e-05
],
"_best_expr": [
"add(u_xx, mul(u,u_x))"
],
"_n_candidates": [
1
]
},
"lhs_label": "u_t",
"equation": {
"form": "EVOLUTION",
"lhs_spec": {
"field": "u",
"axis": "t",
"order": 1
},
"terms": [
[
"u",
{
"kind": "Scalar",
"value": 0.0
}
],
[
"u_x",
{
"kind": "Scalar",
"value": 0.0
}
],
[
"u_xx",
{
"kind": "Scalar",
"value": 0.10018764586954051
}
],
[
"mul(u,u_x)",
{
"kind": "Scalar",
"value": -1.0000022081692426
}
]
],
"attrs": null,
"active_indices": [
2,
3
]
},
"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",
"mul(u,u_x)"
],
"artifacts": null,
"resumed": false,
"resume_source": null
},
"run_record": {
"schema_version": 2,
"evidence_hash_scheme": "kd-evidence-v1",
"created_at": "2026-09-14T04:33:54+00:00",
"cost": {
"wallclock_seconds": 2.295153263024986,
"search_seconds": 2.286355511052534,
"boundary_results": 1,
"boundary_invalid_results": 0,
"preprocessing_seconds": 0.008797751972451806,
"surrogate_train_seconds": null,
"cpu_seconds": 2.2585239999999995,
"tokens_in": null,
"tokens_out": null,
"tokens_cached": null,
"api_cost_usd": null
},
"evidence": {
"instrument": "pysindy",
"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": "add(u_xx, mul(u,u_x))",
"score_kind": "NMSE",
"score_direction": "min",
"headline_coefficient_source": "native",
"catalog_fit": {
"form": "EVOLUTION",
"lhs_spec": {
"field": "u",
"axis": "t",
"order": 1
},
"terms": [
[
"u",
{
"kind": "Scalar",
"value": 0.0
}
],
[
"u_x",
{
"kind": "Scalar",
"value": 0.0
}
],
[
"u_xx",
{
"kind": "Scalar",
"value": 0.10018764586954051
}
],
[
"mul(u,u_x)",
{
"kind": "Scalar",
"value": -1.0000022081692426
}
]
],
"attrs": null,
"active_indices": [
2,
3
]
},
"support": [
"u_xx",
"mul(u,u_x)"
],
"coefficients": [
0.10018764586954051,
-1.0000022081692426
],
"complexity": 2,
"mse": 7.631660410276492e-08,
"nmse": 1.1178326915795506e-05,
"r2": 0.9999888216730842,
"score": 1.1178326915795506e-05,
"residual_stats": {
"mean": -1.3669638288923916e-09,
"std": 0.0002762572845603881,
"max_abs": 0.0038285049487423884,
"n": 51456
},
"invalid_reason": null,
"error_detail": null
},
"evidence_hash": "5c6fd84428860261db401eb095a2fcfee906d67241fca41e9dd90c602b0e9242",
"run_spec": {
"kd_version": "0.7.5",
"config": {
"algorithm": "pysindy",
"extra_optimizer_kwargs": null,
"library_fingerprint": "017dbdceae9f468d",
"max_iter": 20,
"normalize_columns": false,
"provider_kind": "finite_diff",
"seed": 0,
"terms": [
"u",
"u_x",
"u_xx",
"mul(u,u_x)"
],
"threshold": 0.05,
"unbias": true
},
"dataset_cache_fingerprint": "burgers:grid:u:t:lhs_order=1:x,t:periodic=t=N,x=N_u(256, 201)_fb43be53",
"library_fingerprint": "017dbdceae9f468d",
"artifacts": null,
"config_canon_scheme": "kd-config-v1"
},
"run_spec_hash": "5877d8d214e6f610942d5e74ae4b71f27128bb8a84327b4a2330ab255bdf9baf",
"run_spec_hash_scheme": "kd-runspec-v1",
"record_hash": "97fd9f89ca37ca9c4c48814d8ee97ce62a1e97588a0b8a4f8a3d2c8a407c6aa0",
"record_hash_scheme": "kd-record-v1"
},
"score_kind": "NMSE",
"score_direction": "min"
}