# Linear PDE solution with constraints

Consider the following linear PDE:

$$\nabla_q V(q) - M_d(q)M^{-1}(q)\nabla_q V_d(q) = 0,$$

where $$V(q)$$ and $$M(q)$$ are known and $$M_d(q)$$ is a grey box function (e.x., $$M_d(q)$$ is fitted using a neural network). The goal is to find $$V_d(q)$$.

I am able to find a solution for $$V_d(q)$$ using physics informed neural networks. But I need to add some constraints on $$V_d(q)$$, e.g., $$V_d(q)$$ is quasi-convex with a global minimum at $$q^*$$. Instead of using a simple MLP for $$V_d(q)$$, I also tried Input Convex Neural Networks, Input Invex Neural Networks, and hand-crafted loss for the constraints. In all of the cases, the neural network approach does not perform well, either because the constraints are not satisfied or the PDE is not solved.

I also wonder if I can find a solution numerically since all of the other components of this PDE are essentially known and it is just a simple linear PDE.

What are your thoughts on this?

# Edit

I was reluctant to post the whole problem here because I wanted to keep the question simple. However, it seems that additional details may help straighten things up. The PDE above comes from this paper, which is a method for stabilizing a class of underactuated mechanical systems. In the development of the controller, the following nonlinear PDEs need to be solved, $$G^\perp\left(\nabla_q H - M_d M^{-1}\nabla_q H_d + J_2 M_d^{-1} p\right) = 0, \quad (2.9)$$ where $$G^\perp$$ is a full rank left annihilator of the control input matrix $$G$$. In (2.9), $$M_d(q)$$ and $$V_d(q)$$ are unknown, and $$J_2(q, p)$$ is a free parameter. The solution to (2.9) may seem complicated but it can be separated into terms that depend on $$p$$ and terms that are independent of $$p$$ as follows, $$G^\perp\left[\nabla_q(p^\top M^{-1} p) - M_d M^{-1}\nabla_q(p^\top M_d^{-1} p) + 2 J_2 M_d^{-1} p\right] = 0, \quad (2.11)$$ $$G^\perp\left(\nabla_q V - M_d M^{-1}\nabla_q V_d\right) = 0. \quad (2.12)$$ My understanding is that though the solutions of (2.11) and (2.12) also satisfies (2.9), the reverse is not true. Note that (2.11) is still a nonlinear PDE and (2.12) is a linear PDE. So supposedly (2.11) is harder to solve, which is also noted by the authors. Moreover, the authors managed to solve (2.11) and (2.12) with constraints on $$V_d$$ in the examples presented by the papers. So at least in these simple situations, (2.11) and (2.12) can indeed be solved and I believe the solutions are not unique (due to the fact that $$J_2$$ is a free parameter.

My problem is the following. I wanted to solve these PDEs using Neural Networks. From my experiments, (2.9) can be easily solved if there's no constraint on $$V_d$$. But we need the constraint on $$V_d$$ in order for this whole method to work. And by introducing some new cost to the loss, I couldn't get a satisfactory solution for $$V_d$$.

Now, I also tried to solve (2.11) using NN and I can always find a solution since there's not extra constraint. But substituting this obtained $$M_d$$ into (2.12) and try to solve for $$V_d$$ with constraint does not work. In fact, even without adding constraints to $$V_d$$, (2.12) still cannot be solved with an obtained "correct" $$M_d$$ from solving (2.11). I think it makes sense because the co-solution of (2.11) and (2.12) is only a subset to the solution of (2.9).

• PDEs have solutions, and you can only impose additional constraints if the PDE has multiple solutions. On the other hand, if the PDE has only a single solution, then you cannot impose additional constraints: the solution is what it is. So I would first like to ask why you think that there are multiple solutions of the PDE? In fact, what do you know about solutions of the PDE you show to begin with? Nov 25, 2022 at 20:30
• @WolfgangBangerth Thanks for the comment. I have added more details for some background information. See what's below "Edit" in the original post. I think you are raising a good question. (2.12) alone may indeed not have a solution, even $M_d$ satisfies (2.11). However, I think there should at least be one solution when we consider (2.11) and (2.12) together (at least for the numerical examples considered in this paper), and the authors of the paper have provided analytical solutions for the solution of (2.11) and (2.12).
– Evan
Nov 28, 2022 at 21:14
• A summary of my current situation is as follows. Only (2.9) can be solved using NN without constraints on $V_d$ but that's meaningless. (2.11) and (2.12) cannot be solved together, even without constraints on $V_d$. (2.11) and (2.12) cannot be solved in sequence (i.e., solving (2.11) for $M_d$, then use the obtained $M_d$ to solve (2.12)), even without constraints on $V_d$.
– Evan
Nov 28, 2022 at 21:20