Bidirectional dynamic neural networks with physical analyzability

Abstract

The rapid growth in research exploiting deep learning to predict mechanical systems has revealed a new route for system identification; however, the analytic model as a white box has not been replaced in applications because of its open physical information. In contrast, the models generated by end-to-end learning usually lack the ability of physical analysis, which makes them inapplicable in many situations. Consequently, high-accuracy modeling with physical analyzability becomes a necessity. In this paper, we introduce bidirectional dynamic neural networks, a deep learning framework that can infer the dynamics of physical systems from control signals and observed state trajectories. Based on forward dynamics, we train the neural ordinary differential equations in a trajectory backtracking algorithm. With the trained model, the inverse dynamics can be calculated and based on $\textit{Lagrangian}$ $\textit{Mechanics}$, the physical parameters of the mechanical system can be estimated, including inertia, Coriolis and centrifugal forces, and gravity. As a result, the model can seamlessly incorporate prior knowledge, learn unknown dynamics without human intervention, and provide information as transparent as analytic models. We demonstrate our method on simulated 2-axis and 6-axis robots to evaluate model accuracy, including physical parameters and verified its applicability on real 7-axis robots. The experimental results show that this method is superior to the existing methods. This framework provides a new idea for system identification by providing interpretable, physically consistent models for physical systems.

Appendices

Appendix A: The key steps in backpropagation of BDNN

Define $\varvec{\theta }^{(1)}$, $\varvec{\theta }^{(2)}$ and $\varvec{\varepsilon }^{(1)}$, $\varvec{\varepsilon }^{(2)}$ as weights of input and output layer of H and R net in BDNN, define ${\varvec{i}}^{(1)}$ and ${\varvec{i}}^{(2)}$ as the inputs of hidden and output layer neurons, ${\varvec{o}}^{(1)}$ and ${\varvec{o}}^{(2)}$ as inputs of the whole net and outputs of the hidden neurons respectively. In addition, define $\varphi$ as the activation function. Then the key to Algorithm 1 is $\frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial {\varvec{x}}^\top }$, ${\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\theta }}$ and ${\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\varepsilon }}$. According to Eq. (2), the gradient of f to ${\varvec{x}}$ can be derived as:

$$\begin{aligned} \begin{aligned} \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial {\varvec{x}}^\top }&{=}\frac{\partial ({\textbf{H}}({\varvec{u}}{-}{\textbf{R}})) }{\partial {\varvec{x}}^\top }{=}\frac{\partial {\textbf{H}}}{\partial {\varvec{x}}^\top }({\varvec{u}}{-}{\textbf{R}}){-}{\textbf{H}}\frac{\partial {\textbf{R}}}{\partial {\varvec{x}}^\top }\\&=\begin{bmatrix} \frac{\partial {\textbf{H}}}{\partial {\varvec{q}}^\top }({\varvec{u}}-{\textbf{R}})&{\varvec{0}} \end{bmatrix}-{\textbf{H}}\frac{\partial {\textbf{R}}}{\partial {\varvec{x}}^\top } \end{aligned}\nonumber \\ \end{aligned}$$

(11)

The backpropagation for computing $\frac{\partial {\textbf{H}}}{\partial {\varvec{q}}^\top }({\varvec{u}}-{\textbf{R}})$ with function ${\mathscr {D}}$ has been showed in Eq. (4).

The adjoint sensitivity in neural ODEs is expressed as ${\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\theta }}$ and ${\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\varepsilon }}$. For H net in BDNN, the gradient related to the weight of the output layer is directly determined by function ${\mathscr {P}}^{-1}$:

$$\begin{aligned} \begin{aligned} {\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\theta }^{(2)}}&={\mathscr {A}}^\top \frac{\partial {\textbf{H}}}{\partial \varvec{\theta }^{(2)}}({\varvec{u}}-{\textbf{R}})\\&={\mathscr {P}}^{-1}({\mathscr {A}}_{s} \cdot ({\varvec{u}}-{\textbf{R}}) )^\top \odot \varphi ^{'}({\varvec{i}}_{H}^{(2)}){\varvec{o}}_{H}^{(2)\top }\\&=\varvec{\delta }_{H}^{(2)}{\varvec{o}}_{H}^{(2)\top } \end{aligned} \end{aligned}$$

(12)

Note that ${\mathscr {A}}_{s}$ represents the N.th to $2 \times N.th$ elements of adjoint sensitivity ${\mathscr {A}}$. Thus for the hidden layer we have:

$$\begin{aligned} {\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\theta }^{(1)}} =\varvec{\theta }_{nb}^{(2)\top }\varvec{\delta }_{H}^{(2)}\odot \varphi ^{'}({\varvec{i}}_{H}^{(1)}){\varvec{o}}_{H}^{(1)\top } \end{aligned}$$

(13)

Where $\varvec{\theta }_{H,nb}^{(2)}$ means the weight matrix without elements related to bias, and ${\varvec{i}}$ and ${\varvec{o}}$ with subscript H means elements of the tensor produced by H net. Similar expressions are also used in R net:

$$\begin{aligned} \begin{aligned} {\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\varepsilon }^{(2)}}&=-{\mathscr {A}}_{s}^\top {\textbf{H}} \frac{\partial {\textbf{R}}}{\partial \varvec{\varepsilon }^{(2)}} =-{\textbf{H}}{\mathscr {A}}_{s} \odot \varphi ^{'}({\varvec{i}}_{R}^{(2)}){\varvec{o}}_{R}^{(2)\top }\\&=\varvec{\delta }_{R}^{(2)}{\varvec{o}}_{R}^{(2)\top } \end{aligned} \end{aligned}$$

(14)

and:

$$\begin{aligned} {\mathscr {A}}^\top \frac{\partial f({\varvec{x}},{\varvec{u}})}{\partial \varvec{\varepsilon }^{(1)}} =\varvec{\varepsilon }_{nb}^{(2)\top }\varvec{\delta }_{R}^{(2)}\odot \varphi ^{'}({\varvec{i}}_{R}^{(1)}){\varvec{o}}_{R}^{(1)\top } \end{aligned}$$

(15)

Appendix B: Gradients

State estimation is considered as a constrained optimization problem in BDNN:

$$\begin{aligned}&\displaystyle \min _{{\varvec{w}}} L({\varvec{x}}): \ L({\varvec{x}})=\int _{0}^{T}l({\varvec{x}}_t)dt \nonumber \\&\displaystyle \mathrm {s.t.} \ \varvec{\dot{x}}_t-f({\varvec{x}}_t,{\varvec{u}}_t)=0 \\&\displaystyle {\varvec{x}}_t\Big |_{t=0}={\varvec{x}}_0\nonumber \end{aligned}$$

(16)

Define ${\varvec{w}}\in {\{\varvec{\theta },\varvec{\varepsilon }\}}$. According to NODE, the Gradients of BDNN can be derived using $\textit{Lagrange Multiplier}$:

$$\begin{aligned}{} & {} \min _{{\varvec{w}}} {\mathscr {J}}: \ {\mathscr {J}}=\int _{0}^{T}\{l({\varvec{x}}_t)+\varvec{\alpha }_{t}^\top [\varvec{\dot{x}}_t-f({\varvec{x}}_t,{\varvec{u}}_t)]\nonumber \\{} & {} \quad \quad \quad \quad \quad \quad \quad \quad \;\; +\varvec{\mu }^\top {\varvec{x}}_0\}dt \end{aligned}$$

(17)

Then the gradient is:

$$\begin{aligned} \begin{aligned} \frac{dL}{d{\varvec{w}}}&=\frac{d{\mathscr {J}}}{d{\varvec{w}}} \\&=\int _{0}^{T}\{[\frac{dl({\varvec{x}}_t)}{d{\varvec{x}}_t^\top }-\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{x}}_t^\top }-\varvec{{\dot{\alpha }}}_{t}^\top ]\frac{d{\varvec{x}}_t}{d{\varvec{w}}}\\&\quad \;\;-\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}}\}dt \\&\quad \;\;+\varvec{\alpha }_{t}^\top \frac{d{\varvec{x}}_t}{d{\varvec{w}}}\Big |_{t=T}+(\varvec{\mu }^\top -\varvec{\alpha }_{t}^\top )\frac{d{\varvec{x}}_t}{d{\varvec{w}}}\Big |_{t=0} \end{aligned} \end{aligned}$$

(18)

There is obviously a solution:

$$\begin{aligned} \begin{aligned} \varvec{\alpha }_{0}&=\varvec{\mu } \\ \varvec{\alpha }_{T}&=0 \\ \varvec{{\dot{\alpha }}}_{t}^\top&=\frac{dl({\varvec{x}}_t)}{d{\varvec{x}}_t^\top }-\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{x}}_t^\top } \\ \frac{dL}{d{\varvec{w}}}&=-\int _{0}^{T}\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}}dt \end{aligned} \end{aligned}$$

(19)

Let:

$$\begin{aligned} {\varvec{G}}_t=-\int _{t}^{T}\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}}dt \end{aligned}$$

(20)

we have:

$$\begin{aligned} \varvec{\dot{G}}_t=\frac{d}{dt}\Big (\int _{T}^{t}\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}}dt\Big )=\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}}\nonumber \\ \end{aligned}$$

(21)

then there is the gradient of Algorithm 1:

$$\begin{aligned} \begin{aligned} {\varvec{G}}_T&=0 \\ \varvec{\dot{G}}_t&=\varvec{\alpha }_{t}^\top \frac{df({\varvec{x}}_t,{\varvec{u}}_t)}{d{\varvec{w}}} \\ \frac{dL}{d{\varvec{w}}}&={\varvec{G}}_0 \end{aligned} \end{aligned}$$

(22)