Let $\left(M^n, g\right)$ be a closed Riemannian manifold with $C^1$-smooth $g_{i j}$. The spectrum of the Laplace operator on $M$ is discrete. There is a sequence of eigenvalues
$$ 0=\lambda_1<\lambda_2 \leq \lambda_3 \ldots $$
that tend to $\infty$ and a sequence of (real) eigenfunctions $u_k$ such that
$$ \Delta_g u_k+\lambda_k u_k=0 . $$
Our enumeration of eigenvalues is non-standard. We start with $\lambda_1=0$ and $u_1=1$ on $M$. The nodal domains of $u_k$ are the connected components of $M \backslash Z_{u_k}$, where $Z_{u_k}$ is the zero set of $u_k\left(Z_{u_k}\right.$ is called the nodal set of $\left.u_k\right)$. The Courant nodal domain theorem states that the $k$-th eigenfunction $u_k$ has at most $k$ nodal domains. If the multiplicity of an eigenvalue is more than 1 , one may enumerate the eigenfunctions corresponding to this eigenvalue in any order. Our main result is the local version of Courant’s theorem.
In many mathematical and engineering problems, we are interested in finding solutions that satisfy certain constraints. A powerful modern paradigm is to train a neural network as a generator that proposes candidate solutions, and use a differentiable validator (i.e., loss function) to evaluate how well they satisfy those constraints. This feedback is then used to update the network via gradient descent.
In this note, we illustrate this approach by using a neural network to generate vectors that nearly achieve equality in the Cauchy-Schwarz inequality.
1. Problem Setup: Making Cauchy-Schwarz Nearly Tight
Equality holds if and only if x\mathbf{x} and y\mathbf{y} are linearly dependent: y=kx\mathbf{y} = k \mathbf{x} for some scalar kk.
Objective:
Given an input vector x\mathbf{x}, train a neural network NN to output a vector y=N(x)\mathbf{y} = N(\mathbf{x}) such that x,y\mathbf{x}, \mathbf{y} are as close to colinear as possible.
2. Generator: Neural Network Design
Let N(⋅)N(\cdot) be a feedforward neural network (e.g. MLP) with:
Structure: Simple MLP with 1–2 hidden layers (ReLU), and a linear output layer (no activation)
3. Validator: Loss Function Design
To measure how close x,y\mathbf{x}, \mathbf{y} are to colinearity, use cosine similarity: cos(θ)=⟨x,y⟩∥x∥⋅∥y∥+ε\cos(\theta) = \frac{\langle \mathbf{x}, \mathbf{y} \rangle}{\|\mathbf{x}\| \cdot \|\mathbf{y}\| + \varepsilon}
We define the loss as: L(x,y)=1−∣⟨x,y⟩∥x∥⋅∥y∥+ε∣L(\mathbf{x}, \mathbf{y}) = 1 – \left|\frac{\langle \mathbf{x}, \mathbf{y} \rangle}{\|\mathbf{x}\| \cdot \|\mathbf{y}\| + \varepsilon}\right|
L=0L = 0 when x\mathbf{x} and y\mathbf{y} are perfectly aligned or anti-aligned
ε≪1\varepsilon \ll 1 is a small constant for numerical stability
This validator provides a differentiable measure of alignment quality.
4. Training Procedure
Data generation: Sample random input vectors x\mathbf{x} from e.g. N(0,I)\mathcal{N}(0, I)
Loss computation: Evaluate L(x,y)L(\mathbf{x}, \mathbf{y})
Backpropagation: Compute ∇θL\nabla_\theta L and update θ\theta using an optimizer (e.g. Adam)
Repeat until convergence
At the end of training, the network learns to generate vectors y\mathbf{y} nearly colinear with x\mathbf{x}, thus making the Cauchy-Schwarz inequality nearly tight.
5. General Framework: Generator + Validator
This method exemplifies a general and powerful pattern in deep learning:
Component
Role
Description
Neural Network NN
Generator / Solver
Maps input (or noise) to a candidate solution
Validator VV
Loss / Constraint Function
Evaluates how well the candidate satisfies the constraints (must be differentiable)
Optimizer
Learning Engine
Uses gradients to update NN so that the solutions improve over time
Functional problems: e.g. finding extremals in calculus of variations
Neural symbolic systems: e.g. generating logic-constrained expressions
Inverse design: input-to-output mappings constrained by physical or mathematical laws
Conclusion
Training a neural network to minimize a differentiable validator is a powerful method to learn constrained solutions. The Cauchy-Schwarz example shows how even classical inequalities can be embedded into a modern optimization loop, potentially aiding in automated reasoning, symbolic learning, or mathematical discovery.
Would you like this exported to PDF with rendered math? Or should I write a minimal PyTorch implementation to match?
