Deep Differential Network:

This package provides an implementation of a Deep Differential Network. This network architecture is a variant of a fully connected network that in addition to computing the function value $f(\mathbf{x}; \theta)$ outputs the network Jacobian w.r.t. the network input $\mathbf{x}$, i.e., $\mathbf{J} = \partial f(\mathbf{x}; \theta) / \partial \mathbf{x}$. The Jacobian can be computed in closed form with machine precision, with minimal computational overhead and for any non-linearity using the chain rule. Within the implementation we extend the computational graph of a fully connected layer with an additional path to compute the partial derivative w.r.t. the previous layer. Chaining these partial derivatives computes the Jacobian. Therefore, the Jacobian is computed using forward differentiation and the network parameters $\theta$ can be learned by backpropagating through the Jacobian using the standard autograd approach.

Let $f(\mathbf{x}; \theta)$ be the deep network with the parameters $\theta$ and the input $\mathbf{x}$, then the Jacobian can be computed using the chain rule via

$$\frac{\partial f(\mathbf{x}; \theta)}{\partial \mathbf{x}} = \frac{\partial f(\mathbf{x}; \theta)}{\partial \mathbf{h}_{N-1}} \frac{\partial \mathbf{h}_{N-1}}{\partial \mathbf{h}_{N-2}} \hspace{5pt} \cdots \hspace{5pt} \frac{\partial \mathbf{h}_1}{\partial \mathbf{x}}.$$

The partial derivative w.r.t. the previous layer can be computed for the fully connected layer, i.e., $\mathbf{h}_i = g(\mathbf{W}^{T}_{i} \mathbf{h}_{i-1} + \mathbf{b})$, via

$$\hspace{50pt} \frac{\partial\mathbf{h}_i}{\partial\mathbf{h}_{i-1}} = \text{diag} \left( g'(\mathbf{W}_{i}^{T} \mathbf{h}_{i-1} + \mathbf{b}_{i}) \right) \mathbf{W}_{i}.$$

with the non-linearity $g(.)$ and the corresponding element-wise gradient $g'(.)$. Using this approach the Jacobian is computed using a single feed- forward pass and adds very limited computational complexity. The figure shows the extended computational graph and the chaining of the partial derivatives for the computation of the Jacobian.

Prior Work:

This Deep Differential Network architecture was introduced in the ICLR 2019 paper Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning. Within this paper the differential network was used to represent the kinetic and potential energy of a rigid body. These energies as well as the Jacobian of the energies were embedded within the Euler-Lagrange differential equation and the the physically plausible network parameters were learned by minimising the residual of this differential equation on recored data.

The deep differential network architecture was used in the following papers:

• Lutter et. al., (2019). Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning,
  International Conference on Learning Representations (ICLR).
• Lutter et. al., (2019). Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems,
  International Conference on Intelligent Robots & Systems (IROS).
• Lutter et. al., (2019). HJB Optimal Feedback Control with Deep Differential Value Functions and Action Constraints,
  Conference on Robot Learning

If you use this implementation within your paper, please cite:

@inproceedings{lutter2019deep,
  author =      "Lutter, M. and  Ritter, C. and  Peters, J.",
  year =        "2019",
  title =       "Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning",
  booktitle =   "International Conference on Learning Representations (ICLR)",
}

Example:

The example scripts 1d_example_diff_net.py & 2d_example_diff_net.py provide an example to train a differential network to approximate the 1d-function $f(x) = sin(x)$ & the 2d-function $f(x, y) = cos(x)sin(y)$ and the respective Jacobian. The models can be either trained from scratch or the pre-trained models can be loaded from the directory folder. Below you can see the learned approximations of both functions and the Jacobian for the ReLu and Softplus activation. Right now the package supports Linear, ReLu, SoftPlus and Cosine activations but other non-linearities can be easily added.

1d Example: $f(x) = sin(x)$

The ReLu differential network and the SoftPlus differential network are able to approximate the functions very accurately. However, only the SoftPlus network yields a smooth Jacobian, whereas the Jacobian of the ReLu network is piecewise constant due to the non-differentiable point of the ReLu activation.

Performance of the Different Models:

	Type 1 Loss / ReLu	Type 1 Loss / SoftPlus	Type 1 Loss / TanH	Type 1 Loss / Cos	Type 2 Loss / ReLu	Type 2 Loss / SoftPlus	Type 2 Loss / TanH	Type 2 Loss / Cos
MSE $f(\mathbf{x})$	9.792e-6	6.909e-6	8.239e-7	1.957e-7	5.077e-7	8.666e-6	7.563e-6	1.746e-8
MSE $\partial f(\mathbf{x}) / \partial \mathbf{x}$	4.514e-4	1.807e-4	3.099e-5	2.189e-6	2.661e-3	1.494e-4	4.345e-4	5.731e-7
MSE $\partial^2 f(\mathbf{x}) / \partial^2 \mathbf{x}$	4.950e-1	5.930e-3	1.625e-3	5.451e-5	4.950e-1	4.519e-3	8.642e-3	1.209e-5

ReLu Differential Network	SoftPlus Differential Network
TanH Differential Network	Cosine Differential Network

2d Example: $f(x, y) = cos(x)sin(y)$

The ReLu differential network and the SoftPlus differential network are able to approximate the functions very accurately. However, only the SoftPlus network yields a smooth Jacobian, whereas the Jacobian of the ReLu network is piecewise constant due to the non-differentiable point of the ReLu activation.

Performance of the Different Models:

	Type 1 Loss / ReLu	Type 1 Loss / SoftPlus	Type 1 Loss / TanH	Type 1 Loss / Cos	Type 2 Loss / ReLu	Type 2 Loss / SoftPlus	Type 2 Loss / TanH	Type 2 Loss / Cos
MSE $f(\mathbf{x})$	4.151e-5	2.851e-6	2.583e-6	2.929e-7	3.293e-5	4.282e-5	8.361e-6	2.118e-6
MSE $\partial f(\mathbf{x}) / \partial \mathbf{x}$	2.014e-3	3.656e-5	1.525e-5	4.344e-6	4.962e-3	8.128e-4	4.349e-4	7.038e-5
MSE $\partial^2 f(\mathbf{x}) / \partial^2 \mathbf{x}$	9.996e-1	1.288e-3	8.535e-4	1.709e-4	9.996e-1	1.468e-2	1.163e-2	1.316e-3

ReLu Differential Network

SoftPlus Differential Network

TanH Differential Network

Cosine Differential Network

Installation:

For installation this python package can be cloned and installed via pip

git clone https://github.com/milutter/deep_differential_network.git deep_differential_network
pip install deep_differential_network

Contact:

If you have any further questions or suggestions, feel free to reach out to me via michael AT robot-learning DOT de