# Unsupervised Deep Learning Algorithm for PDE-based Forward and Inverse Problems

arXiv: https://arxiv.org/abs/1904.05417 · HTML: https://mlsanity.com/paper/1904.05417/

## Key facts

- Cited by 31 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Apr 2019.

## Built on by (most cited first, 30 of 31)

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- [A Neural PDE Solver with Temporal Stencil Modeling](https://mlsanity.com/paper/2302.08105/) (2023)
- [Guiding continuous operator learning through Physics-based boundary constraints](https://mlsanity.com/paper/2212.07477/) (2022)
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- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
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- [Non-equispaced Fourier Neural Solvers for PDEs](https://mlsanity.com/paper/2212.04689/) (2022)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2202.12358/) (2022)
- [Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows](https://mlsanity.com/paper/2210.15799/) (2022)
- [SciML Agents: Write the Solver, Not the Solution](https://mlsanity.com/paper/2509.09936/) (2025)
- [Separable PINN: Mitigating the Curse of Dimensionality in Physics-Informed Neural Networks](https://mlsanity.com/paper/2211.08761/) (2022)
- [Unsupervised Learning of the Total Variation Flow](https://mlsanity.com/paper/2206.04406/) (2022)
- [Survey of Deep Learning Methods for Inverse Problems](https://mlsanity.com/paper/2111.04731/) (2021)
- [STONet: A Neural-Operator-Driven Spatio-temporal Network](https://mlsanity.com/paper/2204.08414/) (2022)
- [NLP Inspired Training Mechanics For Modeling Transient Dynamics](https://mlsanity.com/paper/2211.02716/) (2022)
- [Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networks](https://mlsanity.com/paper/2302.05322/) (2023)
- [GIT-Net: Generalized Integral Transform for Operator Learning](https://mlsanity.com/paper/2312.02450/) (2023)
- [A Differentiable Framework for Full and Phaseless Data Inversion Using Neural Implicit Contrast-Source Representation](https://mlsanity.com/paper/2508.10555/) (2025)
- [Unlearning Noise in PINNs: A Selective Pruning Framework for PDE Inverse Problems](https://mlsanity.com/paper/2602.19967/) (2026)
- [Faster Optimization on Sparse Graphs via Neural Reparametrization](https://mlsanity.com/paper/2205.13624/) (2022)

## Abstract (arXiv)

We propose a neural network-based algorithm for solving forward and inverse problems for partial differential equations in unsupervised fashion. The solution is approximated by a deep neural network which is the minimizer of a cost function, and satisfies the PDE, boundary conditions, and additional regularizations. The method is mesh free and can be easily applied to an arbitrary regular domain. We focus on 2D second order elliptical system with non-constant coefficients, with application to Electrical Impedance Tomography.