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Deep Learning for Symbolic Mathematics

PyTorch original implementation of Deep Learning for Symbolic Mathematics (ICLR 2020).

This repository contains code for:

  • Data generation
    • Functions F with their derivatives f
    • Functions f with their primitives F
      • Forward (FWD)
      • Backward (BWD)
      • Integration by parts (IBP)
    • Ordinary differential equations with their solutions
      • First order (ODE1)
      • Second order (ODE2)
  • Training
    • Half-precision (float16)
    • Multi-GPU
    • Multi-node
  • Evaluation:
    • Greedy decoding
    • Beam search evaluation

We also provide:

  • Datasets
    • Train / Valid / Test sets for all tasks considered in the paper
  • Trained models
    • Models trained with different configurations of training data
  • Notebook
    • An ipython notebook with an interactive demo of the model on function integration


Datasets and Trained Models

We provide datasets for each task considered in the paper:

Dataset #train Link
Integration (FWD) 45M Link
Integration (BWD) 88M Link
Integration (IBP) 23M Link
Differential equations (ODE1) 65M Link
Differential equations (ODE2) 32M Link

We also provide models trained on the above datasets, for integration:

Model training data Accuracy (FWD) Accuracy (BWD) Accuracy (IBP) Link
FWD 97.2% 16.1% 89.2% Link
BWD 31.6% 99.6% 60.0% Link
IBP 55.3% 85.5% 99.3% Link
FWD + BWD 96.8% 99.6% 86.1% Link
BWD + IBP 56.7% 99.5% 98.7% Link
FWD + BWD + IBP 95.6% 99.5% 99.6% Link

and for differential equations:

Model training data Accuracy (ODE1) Accuracy (ODE2) Link
ODE1 97.2% - Link
ODE2 - 88.2% Link

All accuracies above are given using a beam search of size 10. Note that these datasets and models slightly differ from the ones used in the paper.

Data generation

If you want to use your own dataset / generator, it is possible to train a model by generating data on the fly. However, the generation process can take a while, so we recommend to first generate data, and export it into a dataset that can be used for training. This can easily be done by setting --export_data true:

python --export_data true

## main parameters
--batch_size 32
--cpu true
--exp_name prim_bwd_data
--num_workers 20               # number of processes
--tasks prim_bwd               # task (prim_fwd, prim_bwd, prim_ibp, ode1, ode2)
--env_base_seed -1             # generator seed (-1 for random seed)

## generator configuration
--n_variables 1                # number of variables (x, y, z)
--n_coefficients 0             # number of coefficients (a_0, a_1, a_2, ...)
--leaf_probs "0.75,0,0.25,0"   # leaf sampling probabilities
--max_ops 15                   # maximum number of operators (at generation, but can be much longer after derivation)
--max_int 5                    # max value of sampled integers
--positive true                # sign of sampled integers
--max_len 512                  # maximum length of generated equations

## considered operators, with (unnormalized) sampling probabilities
--operators "add:10,sub:3,mul:10,div:5,sqrt:4,pow2:4,pow3:2,pow4:1,pow5:1,ln:4,exp:4,sin:4,cos:4,tan:4,asin:1,acos:1,atan:1,sinh:1,cosh:1,tanh:1,asinh:1,acosh:1,atanh:1"

## other generations parameters can be found in `` and `src/envs/`

Data will be exported in the prefix and infix formats to:

  • ./dumped/prim_bwd_data/EXP_ID/data.prefix
  • ./dumped/prim_bwd_data/EXP_ID/data.infix

data.prefix and data.infix are two parallel files containing the same number of lines, with the same equations written in prefix and infix representations respectively. In these files, each line contains an input (e.g. the function to integrate) and the associated output (e.g. an integral) separated by a tab. In practice, the model only operates on prefix data. The infix data is optional, but more human readable, and can be used for debugging purposes.

Note that some generators are very fast, such as prim_bwd, which only requires to generate a random function and to differentiate it. The others are significantly longer. For instance, the validity of differential equations is checked (symbolically and numerically) after generation, which can be expensive. In our case, we generated the data across a large number of CPUs to create a large training set. For reproducibility, we provide our training / validation / test datasets in the links above. Generators can be made faster by decreasing the timeout generation time in, but this may slightly reduce the set of equations that the generator can produce.

If you generate your own dataset, you will notice that the generator generates a lot of duplicates (which is inevitable if you parallelize the generation). In practice, we remove duplicates using:

cat ./dumped/prim_bwd_data/*/data.prefix \
| awk 'BEGIN{PROCINFO["sorted_in"]="@val_num_desc"}{c[$0]++}END{for (i in c) printf("%i|%s\n",c[i],i)}' \
> data.prefix.counts

The resulting format is the following:

count1|input1_prefix    output1_prefix
count2|input2_prefix    output2_prefix

Where the input and output are separated by a tab, and equations are sorted by counts. This is under this format that data has to be given to the model. The number of counts is not used by the model, but was not removed in case of potential curriculum learning. The last part consists in simply splitting the dataset into training / validation / test sets. This can be done with the script:

# create a valid and a test set of 10k equations
python data.prefix.counts 10000

# remove valid inputs that are in the train
mv data.prefix.counts.valid data.prefix.counts.valid.old
awk -F"[|\t]" 'NR==FNR { lines[$2]=1; next } !($2 in lines)' <(cat data.prefix.counts.train) data.prefix.counts.valid.old \
> data.prefix.counts.valid

# test test inputs that are in the train
mv data.prefix.counts.test data.prefix.counts.test.old
awk -F"[|\t]" 'NR==FNR { lines[$2]=1; next } !($2 in lines)' <(cat data.prefix.counts.train) data.prefix.counts.test.old \
> data.prefix.counts.test


To train a model, you first need data. You can either generate it using the scripts above, or download the data provided in this repository. For instance:

tar -xvf prim_fwd.tar.gz

Once you have a training / validation / test set, you can train using the following command:


## main parameters
--exp_name first_train  # experiment name
--fp16 true --amp 2     # float16 training

## dataset location
--tasks "prim_fwd"                                                    # task
--reload_data "prim_fwd,prim_fwd.train,prim_fwd.valid,prim_fwd.test"  # data location
--reload_size 40000000                                                # training set size

## model parameters
--emb_dim 1024    # model dimension
--n_enc_layers 6  # encoder layers
--n_dec_layers 6  # decoder layers
--n_heads 8       # number of heads

## training parameters
--optimizer "adam,lr=0.0001"             # model optimizer
--batch_size 32                          # batch size
--epoch_size 300000                      # epoch size (number of equations per epoch)
--validation_metrics valid_prim_fwd_acc  # validation metric (when to save the model)

Additional training parameters can be found in


During training, the accuracy on the validation set is measured at the end of each epoch. However, during training, we only compare the model generated output with the solution in the dataset, although the solution may not be unique. For instance, if the input is:

y'' + y = 0

and that the expected solution in the dataset is a_0 * cos(x) + a_1 * sin(x), then if the model generates a_0 * sin(x) + a_1 * cos(x) the output will be considered invalid because it does not exactly match the one of the dataset. To verify the model output, we plug it into the input equation to verify that this is a valid solution. However, manually verifying the model output can take a lot of time, so we only do this at the end of training, by setting --beam_eval true, and using the following command:


## main parameters
--exp_name first_eval     # experiment name
--eval_only true          # evaluation mode (do not load the training set)
--reload_model "fwd.pth"  # model to reload and evaluate

## dataset location
--tasks "prim_fwd"                                                    # task
--reload_data "prim_fwd,prim_fwd.train,prim_fwd.valid,prim_fwd.test"  # data location

--emb_dim 1024    # model dimension
--n_enc_layers 6  # encoder layers
--n_dec_layers 6  # decoder layers
--n_heads 8       # number of heads

## evaluation parameters
--beam_eval true            # beam evaluation (with false, outputs are only compared with dataset solutions)
--beam_size 10              # beam size
--beam_length_penalty 1.0   # beam length penalty (1.0 corresponds to average of log-probs)
--beam_early_stopping 1     # beam early stopping
--eval_verbose 1            # export beam results (set to 2 to evaluate with beam even when greedy was successful)
--eval_verbose_print false  # print detailed evaluation results

Evaluation with beam can take some time, so we recommend to use not-too-large beams (10 is a good value).

Frequently Asked Questions

How can I run experiments on multiple GPUs?

This code supports both multi-GPU and multi-node training, and was tested with up to 128 GPUs. To run an experiment with multiple GPUs on a single machine, simply replace python in the commands above with:

export NGPU=8; python -m torch.distributed.launch --nproc_per_node=$NGPU

The multi-node is automatically handled by SLURM.

How can I use this code to train a model on a new task?

In src/envs/ you will find several functions gen_prim_fwd, gen_prim_bwd, gen_prim_ibp, gen_ode1, gen_ode2 responsible for the generation of the 5 tasks we considered, inside the environment class CharSPEnvironment. If you want to try a new task, you just need to add a new function gen_NEW_TASK to the environment class.

For all the tasks we considered, the input is composed of an equation with a function y which is the function to find. This procedure is compatible both with integration, and differential equations. For instance, in the case of integration, the input will be of the form y' - F where F is the function to integrate. In the case of differentiation, the input will be of the form y - F' where F is the function to differentiate. If the differential equation is y'' + y = 0 the input will simply be y'' + y. At test time, the y function in the input is replaced by the output of the model which is considered valid if the input is evaluated to 0. Based on the task you consider, you may need to update the evaluator in accordingly.


Deep Learning for Symbolic Mathematics (ICLR 2020) - Guillaume Lample * and François Charton *

  title={Deep learning for symbolic mathematics},
  author={Lample, Guillaume and Charton, Fran{\c{c}}ois},
  journal={arXiv preprint arXiv:1912.01412},


See the LICENSE file for more details.


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