| Algorithm | dlga |
|---|---|
| Dataset | burgers |
| Expression | \( u_{t} = - 0.9993 u u_{x} + 0.0999 u_{xx} \) |
| Best DLGA fitness | 0.00346122 |
| R² | 0.999539 |
| NMSE | 0.0004612 |
| Iterations | 40 |
| Early Stopped | False |
| config.algorithm | dlga |
| config.mode | constant |
| config.library | ['u', 'u_x', 'u_xx', 'u_xxx'] |
| config.solver | svd_null_space |
| config.lhs_auto_select | False |
| config.target_lhs_order | 1 |
| config.epsilon | 0.001 |
| config.pop_size | 40 |
| config.seed | 1 |
| config.auto_upgrade_threshold | 0.001 |
| config.max_modules | 5 |
| config.max_module_length | 5 |
| config.partial_prob | 0.6 |
| config.genes_prob | 0.6 |
| config.crossover_rate | 0.8 |
| config.mutation_rate | 0.4 |
| config.add_rate | 0.4 |
| config.delete_rate | 0.5 |
| config.surrogate_hidden_sizes | [50, 50, 50, 50, 50] |
| config.surrogate_activation | sin |
| config.surrogate_lr | 0.001 |
| config.surrogate_max_epochs | 50000 |
| config.surrogate_patience | None |
| config.surrogate_val_ratio | 0.2 |
| config.surrogate_restore_best | True |
| config.provider_kind | autograd |
Read the published equation, compare its term coefficients with ground truth when available, and check the native search-fit NMSE.
Read the best native score reached at each search iteration.
Track active fitted terms among the recorded top-k distinct structures. Targets remain separate; gray columns have no eligible candidates.
Read the median, interquartile range and min-max native scores of the best representatives of the selected distinct structures. These bands describe the top-k selection, not the entire population.
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.
Per-generation MEAN GA fitness (NMSE + epsilon*length; lower is better) over valid individuals, on a LOG y-axis: single blown-up individuals (svd_null_space normalization) push the mean into the thousands and a linear axis flattens the range the search actually works in. Complements the platform 'convergence' plot, which draws the single global best (the GA keeps its best genome, so the per-generation best is that same curve). A generation appears as a gap whenever its mean is non-finite — one inf-fitness individual suffices, not only all-invalid generations.
Distinct candidate-expression count per generation over the whole population. A collapsing count signals premature convergence.
Mean retained-term count (EvaluationResult.complexity — terms kept in the final fitted equation) of valid individuals per generation. NOT the genome gene count that the epsilon fitness penalty acts on: module-internal bloat (more genes per term) moves the penalty but not this curve. All-invalid generations appear as gaps (nothing was measured).
NN_1 surrogate (Xu 2020 §2.2.1) training loss vs epoch (log-y): train and — when a validation split exists — validation MSE, with the best-validation epoch marked. This is the 'deep learning' evidence of the DLGA pipeline — the autograd-derivative source the GA reads from. Absent (No data panel) when the surrogate was pre-trained or not trained by the platform builder.
{
"result_schema_version": 1,
"best_expression": "add(mul(u, u_x), u_xx)",
"best_score": 0.0034612187302319247,
"iterations": 40,
"early_stopped": false,
"final_eval": {
"mse": 3.145766375617651e-06,
"nmse": 0.00046121873023192473,
"r2": 0.999538781269768,
"score": 0.0034612187302319247,
"complexity": 2,
"coefficients": [
-0.9993157889584017,
0.09990418122526246
],
"is_valid": true,
"error_message": "",
"invalid_reason": null,
"selected_indices": [
0,
1
],
"residuals": "[51456 elements]",
"terms": [
"mul(u, u_x)",
"u_xx"
],
"expression": "add(mul(u, u_x), u_xx)",
"lhs_name": "u_t",
"condition_number": null,
"condition_number_computed": false
},
"actual": "[51456 elements]",
"predicted": "[51456 elements]",
"dataset_name": "burgers",
"algorithm_name": "DLGAPlugin",
"config": {
"algorithm": "dlga",
"mode": "constant",
"library": [
"u",
"u_x",
"u_xx",
"u_xxx"
],
"solver": "svd_null_space",
"lhs_auto_select": false,
"target_lhs_order": 1,
"epsilon": 0.001,
"pop_size": 40,
"seed": 1,
"auto_upgrade_threshold": 0.001,
"max_modules": 5,
"max_module_length": 5,
"partial_prob": 0.6,
"genes_prob": 0.6,
"crossover_rate": 0.8,
"mutation_rate": 0.4,
"add_rate": 0.4,
"delete_rate": 0.5,
"surrogate_hidden_sizes": [
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50,
50,
50,
50
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"surrogate_activation": "sin",
"surrogate_lr": 0.001,
"surrogate_max_epochs": 50000,
"surrogate_patience": null,
"surrogate_val_ratio": 0.2,
"surrogate_restore_best": true,
"provider_kind": "autograd"
},
"recorder": {
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"add(add(add(mul(u, u_x), mul(mul(mul(mul(u, u), u_xx), u_xxx), u_xxx)), u_x), mul(u_xx, u_xxx))",
"mul(u, u_x)",
"add(add(mul(mul(mul(u, u_x), u_xx), u_xxx), u_xxx), mul(u, u_x))",
"add(add(mul(mul(mul(u_x, u_xx), u_xxx), u_xxx), u), u_xx)",
"add(add(add(mul(mul(u_x, u_x), u_x), u), u_xx), u_x)",
"u",
"add(mul(mul(u, u), u_x), u)",
"add(add(u, mul(u_x, u_xxx)), mul(u_x, u_x))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u, u_xx), u_xx), mul(u, u_x)), u_x)",
"add(add(add(mul(u, u_x), mul(mul(mul(mul(u, u), u_xx), u_xxx), u_xxx)), u_x), u_xxx)",
"add(add(add(add(mul(u, u_xx), mul(mul(u_x, u_xx), u_xxx)), mul(u, u_x)), mul(mul(mul(mul(u_x, u_x), u_x), u_xx), u_xxx)), mul(mul(mul(u, u_x), u_xx), u_xxx))",
"mul(u, u_x)",
"add(add(add(add(u, mul(mul(mul(mul(u, u), u), u_x), u_xxx)), mul(mul(u, u_x), u_x)), mul(u_xx, u_xxx)), mul(mul(mul(u, u_x), u_xx), u_xxx))",
"add(u, mul(u, u_xxx))",
"u",
"add(add(u, mul(mul(mul(u, u_xx), u_xxx), u_xxx)), mul(u_x, u_xx))",
"add(mul(mul(mul(u, u_x), u_x), u_xxx), u)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u, u_x), mul(mul(mul(mul(u_x, u_x), u_x), u_xx), u_xxx)), mul(mul(mul(u, u_x), u_xx), u_xxx)), u)",
"add(u, mul(u, u_x))",
"u",
"add(u, mul(mul(mul(mul(u, u_x), u_x), u_xx), u_xxx))",
"add(add(mul(mul(u, u_x), u_xxx), u), mul(mul(mul(mul(u, u), u_x), u_x), u_xxx))",
"add(add(add(u, u_xxx), mul(mul(mul(u, u_x), u_x), u_xx)), mul(mul(u_x, u_xx), u_xx))",
"add(add(add(mul(mul(mul(mul(u, u_x), u_x), u_xx), u_xx), mul(mul(mul(u, u_xx), u_xx), u_xxx)), u), u_x)",
"add(add(add(add(mul(u_x, u_x), mul(mul(u_xx, u_xxx), u_xxx)), mul(mul(mul(u_x, u_xx), u_xx), u_xx)), u), u_xxx)",
"add(add(add(add(u, mul(mul(mul(u, u_xx), u_xxx), u_xxx)), mul(mul(u, u_x), u_xx)), mul(u_xxx, u_xxx)), mul(u_xx, u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(mul(mul(mul(u, u_x), u_x), u_xx), u_xx), mul(mul(mul(u, u_xx), u_xx), u_xxx)), u), mul(u, u_x))",
"add(add(mul(u, u_x), mul(mul(mul(mul(u_x, u_x), u_x), u_xx), u_xxx)), mul(mul(mul(u, u_x), u_xx), u_xxx))",
"add(u, mul(mul(u, u_x), u_x))",
"add(add(mul(mul(u, u_xx), u_xxx), u), u_xx)",
"u",
"add(u, mul(mul(u_x, u_x), u_xx))",
"add(mul(u_xx, u_xxx), u)",
"add(add(mul(mul(u, u_x), u_xxx), u), u_xxx)",
"add(add(mul(mul(u, u_x), u_xxx), u), mul(mul(mul(u_x, u_x), u_xxx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(mul(mul(mul(u, u_x), u_x), u_xx), u_xx), mul(mul(mul(u, u_xx), u_xx), u_xxx)), u_x), mul(u, u_x))",
"add(add(mul(mul(u, u_xx), u_xxx), u), u_xx)",
"u",
"add(u_xxx, u)",
"add(mul(u_xx, u_xxx), u)",
"add(add(mul(u_x, u_xx), mul(mul(mul(mul(u_x, u_x), u_x), u_xx), u_xxx)), u)",
"add(mul(mul(u, u_x), u_xxx), u)",
"add(add(mul(u, u_xx), mul(u_xx, u_xxx)), u)",
"add(u, mul(mul(mul(u, u_x), u_x), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(mul(mul(mul(u_x, u_xx), u_xx), u_xx), mul(u, u_x))",
"add(add(mul(mul(u, u), u_xx), u), u_x)",
"u",
"add(u, mul(u_x, u_xxx))",
"add(mul(u_xx, u_xx), u)",
"add(mul(mul(u, u_xx), u_xxx), u)",
"add(add(mul(mul(mul(mul(u, u_x), u_x), u_xx), u_xxx), u), u_x)",
"add(add(mul(mul(u, u_x), u_xxx), mul(mul(mul(mul(u_x, u_x), u_x), u_xx), u_xxx)), u)",
"add(add(add(add(mul(u_x, u_x), mul(mul(u_xx, u_xxx), u_xxx)), mul(mul(u_xx, u_xx), u_xx)), u), u_xxx)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(mul(u, u_x), mul(mul(u_xx, u_xxx), u_xxx)), mul(mul(u_xx, u_xx), u_xx)), u), u_xxx)",
"add(u, mul(u, u_xxx))",
"u",
"add(add(add(add(mul(mul(mul(u, u_xx), u_xxx), u_xxx), u), mul(mul(mul(u_x, u_xx), u_xx), u_xxx)), mul(mul(mul(u_x, u_x), u_xx), u_xx)), u_xx)",
"add(add(mul(mul(u, u_x), u_xxx), mul(mul(mul(mul(u_x, u_x), u_xx), u_xx), u_xxx)), u)",
"mul(mul(u, u), u)",
"mul(mul(u, u_x), u_x)",
"u_xx",
"u_x"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u, u_x), mul(mul(u_xx, u_xxx), u_xxx)), mul(mul(u_xx, u_xx), u_xx)), u)",
"mul(u, u_x)",
"add(mul(u, u_xx), mul(u, u_x))",
"add(add(add(u_xx, mul(mul(u, u_xx), u_xx)), u_x), u)",
"add(add(add(mul(mul(mul(u, u_xx), u_xxx), u_xxx), u), mul(mul(mul(u_x, u_xx), u_xx), u_xxx)), u_xx)",
"u",
"add(add(add(add(mul(u_xx, u_xxx), u), mul(mul(mul(u, u_x), u_xx), u_xx)), u_xxx), mul(u, u_xx))",
"add(add(mul(mul(mul(mul(u_x, u_x), u_xx), u_xxx), u_xxx), u), mul(mul(mul(mul(u, u_x), u_xx), u_xxx), u_xxx))",
"mul(mul(u, u), u)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(mul(u, u_x), mul(mul(u_xx, u_xxx), u_xxx)), u)",
"mul(u, u_x)",
"add(u_xx, u)",
"add(add(mul(mul(mul(u, u), u_x), u_xx), u), mul(u_xx, u_xxx))",
"u",
"add(add(u_xxx, mul(u_xxx, u_xxx)), u)",
"add(add(u, mul(mul(mul(u, u_x), u_xx), u_xx)), u_xxx)",
"add(u, mul(mul(mul(u, u_xx), u_xx), u_xxx))",
"add(u, mul(mul(mul(mul(u, u_x), u_xx), u_xx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"mul(u, u_x)",
"add(u, u_xx)",
"add(add(mul(mul(mul(u, u), u_x), u_xx), u), mul(u_xx, u_xx))",
"u",
"add(u_xxx, u)",
"add(mul(mul(u_xx, u_xxx), u_xxx), u)",
"add(add(u_xxx, u), mul(u, u_xx))",
"add(add(u, mul(mul(mul(u, u_x), u_xx), u_xx)), u_xxx)",
"add(mul(mul(mul(u, u_xx), u_xxx), u_xxx), u)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(mul(u, u_x), u_xx), mul(mul(mul(u, u_x), u_xxx), u_xxx))",
"add(add(add(mul(mul(u, u), u_xx), mul(mul(u_x, u_x), u_xx)), u), mul(u, u_xxx))",
"add(add(mul(u, u_x), mul(u_xx, u_xxx)), mul(u_xx, u_xx))",
"add(u, u_xx)",
"add(add(u, mul(u_x, u_xx)), u_x)",
"u",
"add(mul(mul(u_x, u_x), u_xx), u)",
"add(add(u_xxx, u), mul(mul(mul(mul(u, u), u), u_x), u_xxx))",
"add(add(u, mul(u_xx, u_xxx)), u_xxx)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"mul(u, u_x)",
"u",
"add(u, u_x)",
"add(u, mul(u_xx, u_xxx))",
"add(add(mul(mul(u, u_xxx), u_xxx), u), u_xxx)",
"add(add(u_xxx, u), mul(mul(mul(u, u_x), u_xx), u_xx))",
"add(add(mul(u_xx, u_xx), mul(mul(u_x, u_x), u_xxx)), u)",
"add(add(add(add(mul(mul(u_x, u_xxx), u_xxx), mul(u, u_xx)), mul(u, u_xxx)), u), u_xxx)",
"add(add(u, mul(u_xx, u_xxx)), mul(mul(mul(mul(u, u_x), u_xx), u_xx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(mul(mul(u_x, u_x), u_x), mul(u, u_x)), mul(mul(mul(mul(u, u), u), u), u_xx))",
"u",
"add(u, u_xxx)",
"add(add(mul(mul(u_x, u_xxx), u_xxx), mul(u, u_xx)), u)",
"add(add(add(add(mul(mul(mul(mul(u_x, u_x), u_x), u_xxx), u_xxx), mul(u_x, u_x)), u), u_xxx), mul(u_xx, u_xxx))",
"mul(mul(u, u), u)",
"u_xx",
"u_xxx",
"mul(u_xx, u_xxx)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(u, mul(u, u_x)), mul(mul(mul(mul(u, u), u), u), u_xx))",
"add(add(add(add(u_x, mul(u, u_xx)), mul(u, u_x)), mul(mul(mul(mul(u, u), u), u), u_xx)), u_xxx)",
"add(u_xx, u)",
"add(add(add(u, u_x), u_xx), mul(mul(mul(u, u_x), u_x), u_xx))",
"u",
"add(u_x, u)",
"add(u_xxx, u)",
"add(mul(mul(mul(mul(u, u_xx), u_xxx), u_xxx), u_xxx), u)",
"add(add(add(add(u_xx, mul(u_xx, u_xxx)), u), mul(mul(mul(mul(u_xx, u_xx), u_xx), u_xxx), u_xxx)), mul(mul(u, u_x), u_x))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(u_x, mul(u, u_xx)), mul(u, u_x)), mul(mul(mul(mul(u, u), u), u), u_xx)), u_xxx)",
"mul(u, u_x)",
"add(add(u, mul(mul(mul(mul(u, u), u), u), u_xx)), mul(u_x, u_xx))",
"add(add(add(add(u_xx, u_xxx), u), mul(mul(mul(mul(u_xx, u_xx), u_xx), u_xxx), u_xxx)), mul(mul(u, u_x), u_x))",
"add(u_xx, u)",
"add(add(u, u_xx), mul(mul(mul(u, u_x), u_x), u_xx))",
"add(add(u_xx, u), mul(mul(mul(mul(u_xx, u_xx), u_xxx), u_xxx), u_xxx))",
"u",
"add(u, mul(mul(u_x, u_xx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(mul(mul(mul(u_xx, u_xxx), u_xxx), u_xxx), u_x), u), u_xxx), mul(u, u_x))",
"add(add(mul(u, u_x), mul(u, u_xx)), mul(mul(mul(mul(u, u), u), u), u_xx))",
"add(mul(mul(mul(mul(u, u), u), u), u_xx), u)",
"add(add(u_xx, u_xxx), u)",
"add(add(add(add(u_xx, u), mul(mul(mul(mul(u_xx, u_xx), u_xx), u_xxx), u_xxx)), mul(mul(u, u_x), u_x)), mul(mul(mul(mul(u, u), u_xx), u_xxx), u_xxx))",
"add(u, mul(u_x, u_xx))",
"u",
"add(u, mul(u_xx, u_xxx))",
"add(u, mul(mul(u_x, u_xx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u, u_x), u), mul(mul(mul(mul(u, u), u), u), u_xx)), mul(mul(u, u_xx), u_xx))",
"add(add(add(add(u, mul(u_x, u_xx)), mul(mul(mul(u, u), u), u_x)), mul(mul(u, u_xx), u_xxx)), mul(mul(mul(u, u), u_x), u_xxx))",
"mul(u, u_x)",
"add(add(mul(u, u_x), u_x), u_xxx)",
"add(add(add(mul(mul(u, u), u_x), u), mul(mul(u, u), u)), mul(mul(u, u), u_xx))",
"add(add(u, mul(u_x, u_xxx)), mul(mul(u, u_x), u_x))",
"add(add(add(u_xx, u), mul(mul(u, u_x), u_x)), mul(mul(mul(mul(u, u), u_xx), u_xxx), u_xxx))",
"add(u, mul(u_x, u_xx))",
"u"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u_x, u_xx), mul(u, u_x)), mul(mul(u, u_xx), u_xxx)), mul(mul(mul(u, u), u_x), u_xxx))",
"mul(u, u_x)",
"add(mul(u, u_x), mul(mul(u_x, u_xx), u_xx))",
"add(add(add(mul(mul(u, u), u_x), u), mul(mul(u, u), u)), mul(mul(u, u), u_xx))",
"add(u, u_xx)",
"u",
"add(mul(mul(u, u_xx), u_xx), u)",
"add(u, u_xxx)",
"add(u, mul(mul(mul(u, u_x), u_xx), u_xx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(u, mul(u, u_x)), mul(mul(u, u_xx), u_xxx)), mul(mul(mul(u, u_x), u_x), u_xxx))",
"mul(u, u_x)",
"add(mul(u, u_x), mul(mul(u_x, u_xx), u_xx))",
"add(add(add(mul(mul(u, u), u_x), u), mul(mul(u, u), u)), mul(mul(u, u), u_xx))",
"add(add(u_xx, u), mul(u_x, u_x))",
"add(add(add(mul(mul(mul(mul(u, u), u_x), u_xx), u_xxx), u_x), u_xx), u)",
"u",
"add(u_x, u)",
"add(u, u_xxx)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(mul(u, u_x), mul(mul(mul(mul(u, u), u_x), u_x), u_xx))",
"add(add(add(u, mul(u, u_x)), mul(mul(u, u_xx), u_xxx)), mul(mul(mul(u, u_x), u_x), u_xxx))",
"add(mul(u, u_xxx), mul(u, u_x))",
"add(add(u_x, mul(u, u_xx)), mul(u, u_x))",
"add(add(mul(u_x, u_xxx), mul(u, u_x)), mul(u, u_xx))",
"add(add(add(add(mul(mul(u, u), u_x), u), mul(mul(u, u), u)), mul(mul(u, u), u_xx)), mul(mul(mul(mul(u, u), u_x), u_xx), u_xx))",
"add(u, mul(u, u_xxx))",
"u",
"add(add(u_x, u), mul(mul(u_x, u_x), u_xx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(u_xx, mul(u, u_x)), mul(mul(mul(mul(u, u_x), u_xx), u_xxx), u_xxx)), u_xxx)",
"add(mul(u, u_x), u)",
"add(add(u, mul(u, u_x)), u_xxx)",
"mul(u, u_x)",
"add(u_x, mul(u, u_x))",
"add(add(add(u_x, mul(u_x, u_xxx)), mul(u, u_x)), mul(u_xxx, u_xxx))",
"add(u_xxx, u)",
"add(add(u_x, u), mul(u, u))",
"add(u, mul(mul(mul(u_x, u_xx), u_xxx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(u, mul(u, u_x)), mul(mul(mul(mul(u_x, u_xx), u_xxx), u_xxx), u_xxx)), u_xxx)",
"mul(u, u_x)",
"add(mul(u, u_x), u_xxx)",
"u",
"add(u, mul(u, u))",
"add(add(u, mul(mul(u, u_x), u_xx)), mul(u, u_xxx))",
"add(add(u_x, u), mul(mul(mul(u_x, u_xx), u_xxx), u_xxx))",
"u_xx",
"u_x"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(mul(u, u_x), mul(mul(mul(mul(u_x, u_xx), u_xxx), u_xxx), u_xxx)), u_xxx)",
"mul(u, u_x)",
"add(mul(u, u_x), u_xxx)",
"add(mul(u, u_x), mul(mul(mul(u, u_x), u_xx), u_xxx))",
"add(add(u, u_xx), u_x)",
"add(add(add(mul(mul(u, u), u_x), u), mul(mul(mul(mul(u, u_x), u_xxx), u_xxx), u_xxx)), u_xx)",
"add(add(add(add(u_xx, mul(mul(mul(mul(u, u_xx), u_xxx), u_xxx), u_xxx)), u), mul(u, u)), u_xxx)",
"add(add(add(add(mul(mul(u_x, u_xx), u_xxx), u), u_xx), mul(mul(mul(mul(u_x, u_xx), u_xx), u_xx), u_xx)), mul(u, u))",
"add(mul(u, u_xxx), u)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(u_x, u_xxx), mul(u, u_x)), mul(u, u_xxx)), mul(u_x, u_xx))",
"add(add(add(mul(u, u_x), mul(mul(mul(mul(u_x, u_xx), u_xxx), u_xxx), u_xxx)), u_xxx), u)",
"mul(u, u_x)",
"add(add(add(mul(mul(u_x, u_xx), u_xxx), u), u_xx), mul(u, u))",
"add(mul(u, u_xxx), u)",
"u",
"add(u, u_x)",
"add(mul(u_xx, u_xxx), u)",
"add(u, mul(u_x, u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(u_xx, mul(u, u_x)), mul(u, u_xxx)), mul(u_x, u_xx))",
"add(add(add(mul(u, u_x), u_xxx), u), mul(u_xx, u_xx))",
"add(add(u_xxx, mul(mul(mul(u, u), u), u_x)), u)",
"mul(u, u_x)",
"u",
"add(u, mul(u, u))",
"add(u, mul(mul(u, u_xx), u_xxx))",
"add(add(mul(mul(mul(mul(u_x, u_x), u_x), u_x), u_xxx), u), mul(mul(mul(mul(u_x, u_x), u_xx), u_xx), u_xxx))",
"u_xx"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(mul(u, u_x), u_xxx), u), u_xx)",
"add(u, mul(u, u_x))",
"add(add(mul(u, u_xxx), u_x), mul(u, u_x))",
"mul(u, u_x)",
"add(add(add(u_xxx, mul(mul(mul(u, u), u_x), u_xxx)), mul(u, u_x)), mul(mul(mul(mul(u_x, u_xx), u_xx), u_xx), u_xxx))",
"add(add(u, mul(u, u)), u_xx)",
"u",
"add(mul(u_xx, u_xxx), u)",
"add(u, mul(mul(mul(mul(u_x, u_xx), u_xx), u_xxx), u_xxx))"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(mul(u, u_x), u_xxx), u), u_xx), mul(mul(u_xx, u_xx), u_xxx))",
"add(add(add(u_xx, mul(mul(mul(u, u), u_x), u_xxx)), mul(u, u_x)), mul(mul(mul(mul(u_x, u_xx), u_xx), u_xx), u_xxx))",
"mul(u, u_x)",
"add(u_x, mul(u, u_x))",
"add(mul(u, u_x), mul(u_x, u_x))",
"add(u, mul(mul(u, u), u_xx))",
"add(u, mul(u_x, u_xx))",
"u",
"add(mul(u_xx, u_xxx), u)"
],
[
"add(add(u, mul(u, u_x)), u_xx)",
"add(add(add(add(u_xx, mul(mul(mul(u, u), u_x), u_xxx)), mul(u, u_x)), mul(mul(mul(mul(u_x, u_xx), u_xx), u_xx), u_xxx)), mul(u, u_xxx))",
"add(add(add(mul(u, u_x), u_xxx), u), mul(mul(u_xx, u_xx), u_xxx))",
"add(mul(u, u_x), u_xxx)",
"add(u, u_xx)",
"add(add(add(mul(u_x, u_x), mul(mul(mul(mul(u, u_x), u_xx), u_xx), u_xxx)), u), u_xx)",
"u",
"add(u, mul(mul(u_x, u_xx), u_xxx))",
"add(add(add(u, mul(mul(u_x, u_x), u_xx)), u_x), mul(mul(u, u_x), u_xx))",
"add(add(add(mul(mul(u_xx, u_xx), u_xxx), u), mul(mul(u, u_xx), u_xx)), mul(u, u))"
],
[
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