This is a general approach to convert a neural network into an analytic equation. The technique works as follows:

1. Encourage sparse latent representations
2. Apply symbolic regression to approximate the transformations between in/latent/out layers
3. Compose the symbolic expressions.

In the [paper](https://arxiv.org/abs/2006.11287), we show that we find the correct known equations, including force laws and Hamiltonians, can be extracted from the neural network. We then apply our method to a non-trivial cosmology example-a detailed dark matter simulation-and discover a new analytic formula which can predict the concentration of dark matter from the mass distribution of nearby cosmic structures. The symbolic expressions extracted from the GNN using our technique also generalized to out-of-distribution data better than the GNN itself. Our approach offers alternative directions for interpreting neural networks and discovering novel physical principles from the representations they learn.

**CPT**, or **Chinese Pre-trained Unbalanced Transformer**, is a pre-trained unbalanced [Transformer](https://paperswithcode.com/method/transformer) for Chinese natural language understanding (NLU) and natural language generation (NLG) tasks. CPT consists of three parts: a shared encoder, an understanding decoder, and a generation decoder. Two specific decoders with a shared encoder are pre-trained with masked language modeling (MLM) and denoising auto-encoding (DAE) tasks, respectively. With the partially shared architecture and multi-task pre-training, CPT can (1) learn specific knowledge of both NLU or NLG tasks with two decoders and (2) be fine-tuned flexibly that fully exploits the potential of the model. Two specific decoders with a shared encoder are pre-trained with masked language modeling (MLM) and denoising auto-encoding (DAE) tasks, respectively. With the partially shared architecture and multi-task pre-training, CPT can (1) learn specific knowledge of both NLU or NLG tasks with two decoders and (2) be fine-tuned flexibly that fully exploits the potential of the model.

Chinese Pre-trained Unbalanced Transformer

CPT: A Pre-Trained Unbalanced Transformer for Both Chinese Language Understanding and Generation

Symbolic Deep Learning

Discovering Symbolic Models from Deep Learning with Inductive Biases

**Online Normalization** is a normalization technique for training deep neural networks. To define Online Normalization. we replace arithmetic averages over the full dataset in with exponentially decaying averages of online samples. The decay factors $\alpha\_{f}$ and $\alpha\_{b}$ for forward and backward passes respectively are hyperparameters for the technique.

We allow incoming samples $x\_{t}$, such as images, to have multiple scalar components and denote
feature-wide mean and variance by $\mu\left(x\_{t}\right)$ and $\sigma^{2}\left(x\_{t}\right)$. The algorithm also applies to outputs of fully connected layers with only one scalar output per feature. In fact, this case simplifies to $\mu\left(x\_{t}\right) = x\_{t}$ and $\sigma\left(x\_{t}\right) = 0$. Denote scalars $\mu\_{t}$ and $\sigma\_{t}$ to denote running estimates of mean and variance across
all samples. The subscript $t$ denotes time steps corresponding to processing new incoming samples.

Online Normalization uses an ongoing process during the forward pass to estimate activation means
and variances. It implements the standard online computation of mean and variance generalized to processing multi-value samples and exponential averaging of sample statistics. The
resulting estimates directly lead to an affine normalization transform.

$$ y\_{t} = \frac{x\_{t} - \mu\_{t-1}}{\sigma\_{t-1}} $$ 

$$ \mu\_{t} = \alpha\_{f}\mu\_{t-1} + \left(1-\alpha\_{f}\right)\mu\left(x\_{t}\right) $$

$$ \sigma^{2}\_{t} = \alpha\_{f}\sigma^{2}\_{t-1} + \left(1-\alpha\_{f}\right)\sigma^{2}\left(x\_{t}\right) + \alpha\_{f}\left(1-\alpha\_{f}\right)\left(\mu\left(x\_{t}\right) - \mu\_{t-1}\right)^{2} $$

Source	Discovering Symbolic Models from Deep Learning with Inductive Biases
Year	2000
Data Source	CC BY-SA - https://paperswithcode.com