Deep Learning for Solving PDEs: Career Prospects and Learning Directions

Deep learning for solving partial differential equations (PDEs) was once regarded as a hot interdisciplinary direction between computational mathematics and AI, but with the rise of large models and changes in the job market, the real prospects of this field need to be re-examined. A researcher in data science at Peking University shares their judgment on the technical nature, academic ecosystem, and career出路 of this field based on their own interview experience.

Technical Barrier: Why Some People Think It's "Simple"

The core idea of deep learning for solving PDEs is not complicated: construct a fully connected neural network, input the independent variables of the equation, output the numerical solution, and train the network to fit the partial differential equation. Compared with traditional computational mathematics, which requires meshing, designing step sizes, and writing C++ or Python code to handle complex fluid dynamics equations, the deep learning approach bypasses these steps.

As mentioned in the video, it "avoids meshing, avoids that weird analysis of errors and time steps," and even avoids convergence analysis.

This "simplicity" is relative—deep learning itself is not easy, but for researchers with a mathematics background, the entry barrier is indeed lower than traditional computational mathematics. Around 2019, papers shared by Professor Weinan E and others made many graduate students feel this path was viable: a simple fully connected network could start training; whether it could be optimized to ideal accuracy was another matter.

The general workflow of traditional computational mathematics for solving PDEs includes: geometric modeling and mesh generation, selection of discretization schemes (finite difference, finite element, finite volume, etc.), handling boundary conditions, time advancement or iterative solving, and post-processing and error analysis. Each step has mature theoretical support and engineering practice, but also implies high implementation complexity. The deep learning method compresses this workflow into "define network—construct loss—optimize training," bypassing mesh generation and scheme design, but the cost is transferring the difficulty to optimization and generalization: network architecture, activation functions, loss weights, and sampling strategies all require repeated trial and error, lacking systematic design guidelines.

Academic Ecosystem: Easy to Publish Low-Quality Papers, Hard to Innovate

This direction was relatively hot between 2019 and 2023, but the video author bluntly points out that it is "particularly suitable for publishing low-quality papers," rather than for innovative research. Most papers are typical low-quality work; a considerable portion do not release source code yet claim excellent results.

The author reminds academic researchers to maintain a skeptical attitude: others' results looking good does not necessarily mean your work is inferior, because their results may also be inflated. The barrier to fraud is invisible—boasting is easy, but debunking requires solid professional knowledge. Taking deep learning for PDEs as an example, insiders might spot issues, but in abstract algebra or harmonic analysis, outsiders cannot judge authenticity at all.

Two Technical Routes: Data-Driven and Physics-Informed

There are mainly two paths for deep learning in solving PDEs. One is data-driven, relying on large amounts of numerical solution data to train the network; the other is physics-informed, most typically PINN (Physics-Informed Neural Network).

The difference between the two lies in whether the equation residual is included as part of the loss function. The physics-informed approach attempts to make the network satisfy equation constraints during training, reducing dependence on data. But regardless of the route, both face the same fundamental problem: convergence cannot be guaranteed before training. Factors such as learning rate and random initialization weights can lead to training failure, and currently few people provide clear convergence analysis—under what prior conditions can a fully connected network converge to the numerical solution of a partial differential equation.

PINN (Physics-Informed Neural Networks) was proposed by George Karniadakis's team at Brown University in 2019. Its core idea is to directly add the PDE residual term to the loss function, so that the network satisfies physical equation constraints while fitting observed data. The loss function is usually a weighted combination of data terms, equation residual terms, and boundary/initial condition terms; the choice of weights itself is a sensitive hyperparameter. The advantage of PINN is that it can be trained even in small-data or label-free scenarios, but the training process is prone to multi-task loss competition, leading to slow convergence or local minima. Subsequent works such as XPINN and cPINN have improved it with domain decomposition and parallelization, but theoretical guarantees of convergence remain scarce.

Accuracy Requirements: The Crux of Industrial Deployment Difficulties

Why does deep learning thrive in computer vision and large language models but struggle to be deployed in PDE solving? The core lies in the difference in accuracy requirements.

Face recognition can determine "whether it's the same person" with 98% or 99% confidence, allowing some error.

Large language models can tolerate slight deviations when answering questions. But numerical solutions of partial differential equations often require errors of 10^-3 to 10^-4; computational mathematics also has error analysis after meshing as a guarantee, whereas deep learning for PDEs lacks a numerical analysis process—whether the result is good or usable often cannot be judged.

A loss function dropping from 10^-2 to 10^-3 may seem like just one order of magnitude, but in reality it's "a slight difference leads to a huge mistake," absolutely unusable in industrial scenarios. The author believes that only when the loss function drops to an extremely low level and the network truly converges to the solution of the equation can the result be used.

In traditional numerical analysis, error sources can be decomposed into truncation error and rounding error, and systematically controlled by mesh refinement (h-refinement) or increasing basis function order (p-refinement). The finite element method has Céa's lemma and a priori/a posteriori error estimates; the finite volume method has stability and convergence theory. These tools allow engineers to give credible error upper bounds for computational results. Deep learning for PDEs lacks a corresponding system: network training is a non-convex optimization process, and the final loss value is only an empirical metric on the training set, which cannot be directly converted into an error bound of the solution on the continuous domain. Even if the loss is small, spectral bias may exist—the network preferentially fits low-frequency components and ignores high-frequency details, causing severe distortion in local regions of the solution.

Employment Reality: Scarce Relevant Positions

During job hunting in 2022-2023, the author found that only a very few institutions such as Microsoft Research Asia and Huawei had relevant positions; other companies had almost none. A senior PhD student who had published several papers eventually got a faculty position, but also could not find a matching job in the corporate market.

At the time, many companies or university professors used this direction to attract investment, but as investors shifted to large models, the popularity of deep learning for PDEs further declined. The author judges that in the next few years, it will only become harder to find relevant jobs in this direction. The development of large models may even directly impact basic mathematics positions—OpenAI once claimed its models can solve complex mathematical problems, and it is not impossible for large models to directly output numerical solutions of fluid dynamics equations in the future.

Advice for Learners

The author's attitude is clear: using deep learning for PDEs code as an exercise for learning Python and introductory deep learning is fine; but do not treat it as a career direction. If you want to find a relevant job, honestly participating in company internships is more practical.

As for the academic path, if you are already pursuing a master's or PhD in this direction, publishing papers and getting a degree is feasible, but remain clear-headed about paper quality and do not be fooled by low-quality results. Regarding the phenomenon of junior master's and PhD students dropping out due to the impact of large models, the author considers this a serious signal, worth everyone carefully considering their direction choices.

This field is not without value—as an intersection of AI and scientific computing, it still has research significance. But viewing "research interest" and "career出路" separately may be a more rational attitude.

One-line Summary

Deep learning for PDEs has a low entry barrier but no convergence guarantee, insufficient accuracy for deployment, scarce jobs, and is not suitable as a career direction.

Paragraph Summary

Based on the interview experience and observations of a data science researcher at Peking University, this article re-examines the real prospects of deep learning for solving partial differential equations. The direction has a lower entry barrier because it bypasses mesh generation and error analysis and was relatively hot from 2019 to 2023, but academic papers are often low-quality, with some claiming excellent results without releasing code. Both data-driven and physics-informed (PINN) routes face the fundamental problem of no convergence guarantee before training. The core obstacle to industrial deployment is accuracy: PDE numerical solutions often require errors of 10^-3 to 10^-4, while deep learning methods lack numerical analysis guarantees, and small differences in loss functions can make results completely unusable. The job market has very few relevant positions, mainly concentrated in a few institutions such as Microsoft Research Asia and Huawei, and investors have shifted to large models. The author suggests using relevant code as an introductory deep learning exercise but not as a career direction; for job hunting, internships should be prioritized; for those already pursuing master's or PhD in this direction, publishing papers and getting a degree is feasible, but they need to remain clear-headed about paper quality.