# Deep Neural Network Approach to Forward-Inverse Problems

arXiv: https://arxiv.org/abs/1907.12925 · HTML: https://mlsanity.com/paper/1907.12925/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2019.

## Built on by (most cited first, 11 of 11)

- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [Enhanced Physics-Informed Neural Networks with Augmented Lagrangian Relaxation Method (AL-PINNs)](https://mlsanity.com/paper/2205.01059/) (2022)
- [A Dimension-Augmented Physics-Informed Neural Network (DaPINN) with High Level Accuracy and Efficiency](https://mlsanity.com/paper/2210.13212/) (2022)
- [Lagrangian dual framework for conservative neural network solutions of kinetic equations](https://mlsanity.com/paper/2106.12147/) (2021)
- [Uncertainty quantification and inverse modeling for subsurface flow in 3D heterogeneous formations using a theory-guided convolutional encoder-decoder network](https://mlsanity.com/paper/2111.08691/) (2021)
- [ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines](https://mlsanity.com/paper/2501.09395/) (2025)
- [Deep Learning Aided Laplace Based Bayesian Inference for Epidemiological Systems](https://mlsanity.com/paper/2210.08865/) (2022)
- [Estimation of System Parameters Including Repeated Cross-Sectional Data through Emulator-Informed Deep Generative Model](https://mlsanity.com/paper/2412.19517/) (2024)
- [Provable wavelet-based neural approximation](https://mlsanity.com/paper/2504.16682/) (2025)
- [Neural Network-Based Parameter Estimation for Non-Autonomous Differential Equations with Discontinuous Signals](https://mlsanity.com/paper/2507.06267/) (2025)
- [Optimal Linear Baseline Models for Scientific Machine Learning](https://mlsanity.com/paper/2508.05831/) (2025)

## Abstract (arXiv)

In this paper, we construct approximated solutions of Differential Equations (DEs) using the Deep Neural Network (DNN). Furthermore, we present an architecture that includes the process of finding model parameters through experimental data, the inverse problem. That is, we provide a unified framework of DNN architecture that approximates an analytic solution and its model parameters simultaneously. The architecture consists of a feed forward DNN with non-linear activation functions depending on DEs, automatic differentiation, reduction of order, and gradient based optimization method. We also prove theoretically that the proposed DNN solution converges to an analytic solution in a suitable function space for fundamental DEs. Finally, we perform numerical experiments to validate the robustness of our simplistic DNN architecture for 1D transport equation, 2D heat equation, 2D wave equation, and the Lotka-Volterra system.