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.

[Khalil-Work]

Short answer: for ODE systems like Lotka–Volterra you usually do not need to enforce conservation explicitly — the conserved quantity is a consequence of the equations, and a PINN that satisfies the ODEs on the whole domain automatically preserves it up to training error. Enforcing it explicitly is worth it when (a) the quantity is the thing you actually care about (e.g. a probability mass, so the solution stays physical), or (b) the residual error accumulates along long time horizons.

Two tested patterns:

1. Soft penalty on the conserved functional (what I would use for probability-style constraints): add a loss term that measures the drift of the conserved quantity from its initial value over the collocation batch. For a global integral constraint like int p dx = 1, the working pattern in stock DeepXDE is a small BC subclass whose collocation points ARE fixed Gauss–Legendre nodes, so its error() can quadrature the network over the domain and return the mismatch (demonstrated in this discussion: Constraining the solution to be a probability density #2027 (comment) — mass error 2.4e-4 on a Fokker–Planck problem vs total collapse without it).

2. Hard structural enforcement: for linear conservation laws like sum_i u_i(t) = const, an output transform can make it exact by construction — e.g. write u_k = g_k - mean(g) - (u_k(0) correction) or parametrize differences only. This removes the drift entirely but is problem-specific.

When NOT to bother: I tested the drift question on a stationary homogeneous problem where the scale is genuinely free — without any anchor the PINN collapses to the trivial scale (mass 0.0003, 99.97% error in the linked test), so there the constraint is essential, not cosmetic. For your ODEs, first train unconstrained and measure the conserved-quantity drift on a held-out dense grid; add the penalty only if the drift matters at your tolerance. Keeping the penalty weight moderate (the constraint is a regularizer) avoids degrading the ODE residual fit.