Conserved quantities in ODEs/PDEs: Should I enforce conservation equations explicitly?

[ululi1970]

ODEs and PDEs usually have quantities that are conserved by the dynamics: For example consider the simple system

$$\left\{\begin{array}{c} \frac{dx}{dt}=v \\ \frac{dv}{dt}=-\frac{dV}{dx} \end{array}\right. (*)$$

if $(x(t),v(t))$ is a solution of this system, then $E(t)=v^2/2+V(x)$ is constant in time with a value $E_0$ determined by the initial condition. To solve this with deepXDE I can (1) just enforce the system (*) in the loss function or (2) I can add an additional term to the loss function $(v^2/2+V(x)-E_0)^2$.

Of course, more complicated systems may have multiple constraints and/or will satisfy additional equations. For example, if in the above system (*) I add a linear "friction" term $-v$ to the rhs of the second equation then the solution satisfies

$$\frac{d}{dt}\left(\frac{v^2}{2}+V(x)\right)=-v^2$$

My question is this: is it better to just solve the "barebone" system, or is it better to add as many additional constrains are known?

[lululxvi]

In general, yes, adding more constraints can help.