🤖 AI Summary
This work addresses the challenge of efficiently and accurately solving parameterized problems—such as parametric ordinary differential equations and inverse problems with wave constraints—in severely ill-posed regimes. The authors propose a gradient flow–based neural network training method that integrates analytic activation functions with a residual network architecture, reformulating parameter learning as the solution of an ordinary differential equation system. By leveraging Łojasiewicz inequality theory, the approach provides rigorous guarantees of convergence during training. Experimental results demonstrate that the method effectively captures the dependency structure of parametric ODE solutions and yields reasonable approximations to inverse problems even under extreme ill-posedness, offering a stable and computationally feasible framework for tackling complex parameterized systems.
📝 Abstract
We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.