Searching for Jiajing Guan usually leads to an applied mathematician whose work sits at the intersection of numerical analysis, partial differential equations, scientific computing, and machine learning. Her public academic record traces a path from undergraduate mathematics research at George Mason University to a PhD in applied mathematics at the University of Maryland, followed by work connected with quantitative research.
What makes this profile worth understanding is not simply a list of degrees or awards. The more interesting story is how classical computational mathematics can be combined with modern neural-network methods to solve difficult physical models. That combination defines much of her publicly documented research.
This guide explains who the researcher is, what she studied, why her work on physics-informed neural networks matters, and how her academic projects fit into the wider development of scientific machine learning.
Who Is Jiajing Guan?
Jiajing Guan is an applied mathematician and researcher known publicly for work involving numerical partial differential equations, dynamical networks, scientific computing, and physics-informed neural networks, commonly called PINNs.
George Mason University records show that she studied mathematics there, received an honorable mention in the Goldwater Scholarship competition, participated in research programs, and later received the 2019 Mary K. Cabell Award for an outstanding mathematics student. The same university record states that she planned to use an NSF Graduate Research Fellowship to pursue a PhD in applied mathematics at the University of Maryland.
The University of Maryland subsequently listed Dr. Jiajing Guan among its Spring 2025 Applied Mathematics, Statistics, and Scientific Computation degree recipients. Her dissertation was titled Resolving Steep Gradients for Physics-Informed Neural Networks: Richards’ Equation and Convection-Diffusion Equation, under the supervision of Howard Elman.
A publicly indexed professional profile later described her as a quantitative researcher with a PhD in applied mathematics. Because employment information can change, that should be treated as a time-sensitive professional description rather than a permanent biographical fact.
Quick Facts
- Field: Applied mathematics
- Core areas: Numerical PDEs, machine learning, scientific computing and dynamical systems
- Undergraduate institution: George Mason University
- Doctoral institution: University of Maryland
- PhD completion: 2025
- Dissertation focus: Improving physics-informed neural networks for equations with steep gradients
- Publicly documented applications: Convection-diffusion equations, Richards’ equation, network reconstruction and reduced-order modeling
Why Jiajing Guan’s Academic Path Matters
The academic development of Jiajing Guan shows a progression that is common among strong computational researchers but difficult to execute well: begin with rigorous mathematics, gain experience in numerical modeling, and then move into machine learning without abandoning the physics or equations that define the original problem.
At George Mason University, her work was already connected with applied questions. A university record identifies her as a participant in research examining links between thermodynamic entropy and information entropy in electroconvecting liquid crystal. Another undergraduate project explored the limits of reconstructing dynamical networks from incomplete and noisy observations.
She also completed numerical-analysis projects involving heat distribution in a cooling fin and audio compression using a modified discrete cosine transform. These projects are educational in scope, but they reveal the practical foundation behind later work: translate a real process into mathematical form, choose a numerical method, test the method, and measure its error.
That foundation matters because machine learning in scientific computing is not only about fitting data. Reliable work requires knowledge of differential equations, discretization, stability, approximation error, boundary conditions, and the physical meaning of a solution.
Jiajing Guan and Physics-Informed Neural Networks
A major part of Jiajing Guan’s doctoral research concerns physics-informed neural networks. PINNs are neural networks trained not only from examples but also from mathematical laws, usually expressed as differential equations.
In a conventional supervised-learning problem, a model learns from many input-output pairs. In a PINN, the differential equation itself contributes to the loss function. The network is penalized when its predicted solution fails to satisfy the governing equation, initial conditions, or boundary conditions.
That idea is attractive because it can reduce dependence on large labeled datasets and provide a flexible, mesh-free approximation framework. Yet PINNs can struggle badly when a physical solution changes sharply over a small spatial region.
The Steep-Gradient Problem
The central challenge addressed in her dissertation is easy to describe, even if it is mathematically difficult.
Imagine a model in which the solution stays almost flat across most of the domain but changes abruptly near one boundary or internal layer. A standard neural network may spend too much of its capacity representing the smooth region and too little resolving the sharp transition.
Traditional numerical solvers address this by refining the mesh or adapting computational resources around the difficult region. A PINN does not automatically know where that extra resolution is needed.
The University of Maryland dissertation record explains that her research examined two important PDE models:
- Richards’ equation, used to describe unsaturated flow through porous media
- The convection-diffusion equation, used to represent processes combining transport and diffusion
Her work proposed architecture changes, loss reformulation, input transformations, and training strategies intended to improve PINN accuracy and robustness when solutions contain steep gradients.
What Jiajing Guan Contributed to PINN Research
The contribution associated with Jiajing Guan is not a claim that PINNs replace classical numerical methods. Her research is more focused: identify where standard PINNs fail, explain part of that failure, and develop targeted strategies that improve their performance.
1. Input Transformations for Known Gradient Locations
When the approximate position of a sharp layer is known, the coordinates supplied to the neural network can be transformed so that the difficult region receives more representational attention.
Her 2024 paper with Howard Elman studied transformed physics-informed neural networks for convection-diffusion equations. The work examined PINNs as tools for correcting oscillatory finite-difference solutions and for modifying reduced solutions of unperturbed problems. It also analyzed why input transformations can improve results through the behavior of neural tangent kernels.
In practical language, an input transformation changes the way the network “sees” the domain. Instead of treating every region as equally simple, the transformation stretches or concentrates coordinates around the area where the solution is hardest to learn.
2. Handling Unknown or Complicated Layer Locations
Known gradient locations are the easier case. In many real problems, the difficult region is not known before solving the equation.
The dissertation introduces an Auxiliary-Input PINN, described as an architecture that adapts spatial transformations using an additional input. It also uses a successive training strategy so the model can learn solutions without being given the gradient position in advance.
This is important because a method that requires perfect prior knowledge has limited value. A more adaptive model can potentially work on problems where sharp transitions move with time, depend on uncertain parameters, or emerge from nonlinear interactions.
3. Causality and Surface Flux in Richards’ Equation
For Richards’ equation, the dissertation describes a framework that incorporates surface flux as an input and uses discrete residuals to enforce causality.
Causality matters in time-dependent models because the predicted state at a later time should develop consistently from earlier states. A training procedure that ignores this order can converge to a numerically convenient answer that does not reflect the actual evolution of the system.
By redesigning inputs and residual terms, the method tries to make the learning process respect more of the problem’s physical and temporal structure.
Earlier Research by Jiajing Guan on Dynamical Networks
Before the doctoral work on PINNs, Jiajing Guan co-authored research on the reconstruction of dynamical networks with Tyrus Berry and Timothy Sauer.
The study introduced an observability condition number for systems whose dynamics are known but whose trajectories are observed at only a subset of network nodes. It examined how observation noise affects the ability to reconstruct the hidden parts of the system.
This problem appears in many settings. A network may represent interacting biological units, sensors, infrastructure, financial variables, or coupled physical components. Researchers often cannot measure every node directly.
The key practical question is therefore not merely, “Can we estimate the missing values?” It is, “When is reliable reconstruction mathematically impossible because the available observations are too weak or too noisy?”
The condition-number approach helps identify which observed nodes are informative and when the reconstruction problem becomes unstable. That type of thinking connects naturally with her later work: both lines of research focus on understanding the limits of a computational method rather than reporting accuracy alone.
Jiajing Guan’s Work on Machine Learning for PDEs
A University of Maryland project completed by Jiajing Guan examined machine-learning methods for parameter-dependent partial differential equations. The project compared a proper orthogonal decomposition neural-network reduced-basis method with physics-informed neural networks.
The study tested how network depth, network structure, and training-sample size affected approximations for problems including an unsteady Burgers equation and a nonlinear diffusion equation. It also identified weaknesses in standard PINNs for singularly perturbed convection-diffusion problems and explored techniques from singular perturbation theory to improve accuracy.
This stage of the research is useful because it shows how a dissertation topic can emerge from systematic experimentation.
A researcher first asks:
- Which methods work on ordinary test problems?
- Where do they fail?
- Is the failure caused by optimization, architecture, sampling, or the mathematical structure of the equation?
- Can established numerical-analysis ideas guide the machine-learning design?
The later transformed-PINN work can be understood as a deeper response to those questions.
Teaching, Service, and Mathematical Community Work
The public record for Jiajing Guan also includes teaching and academic-service activity.
A University of Maryland course page lists her as a teaching assistant and grader for an undergraduate numerical-analysis course in Fall 2020. The course covered core computational topics and MATLAB-based numerical work.
The university’s Women in Math organization also lists Jiajing “JJ” Guan as its webmaster for the 2020–2021 academic year.
These roles do not define a research career, but they help explain the broader academic profile. Teaching numerical analysis strengthens the ability to communicate methods clearly. Service work supports the research community and gives students experience beyond individual publications.
Why This Research Has Wider Scientific Value
The work linked to Jiajing Guan belongs to a wider field often called scientific machine learning. The aim is not to use neural networks simply because they are fashionable. The aim is to combine data-driven approximation with physical laws, mathematical structure, and reliable numerical reasoning.
That approach matters in fields where experiments are costly, observations are incomplete, or classical simulations require enormous computing resources.
Potential application areas include:
- Groundwater and soil-moisture modeling
- Heat and mass transfer
- Fluid transport
- Environmental systems
- Biological processes
- Engineering design
- Reduced-order simulation
- Parameter estimation
- Digital twins
- Quantitative modeling
These applications do not mean that every method in one dissertation is already ready for production. Research typically establishes concepts, algorithms, tests, and limitations before broader deployment.
The durable value lies in the methodology: do not hide model failure—study it. Do not assume a neural network learns the correct physics—enforce and test the governing structure. Do not compare only against an easy baseline—examine difficult regimes where numerical errors become visible.
How to Evaluate Information About Jiajing Guan Online
People searching for Jiajing Guan should be aware that the name can refer to more than one person. Search results include profiles associated with mathematics, biomedical research, athletics, and other fields.
For accurate research, prioritize sources in this order:
- University pages and repositories
- Original papers and preprints
- Official conference or seminar records
- Self-maintained professional profiles
- Third-party people-search websites
Academic repositories are especially valuable because they provide exact titles, advisors, dates, abstracts, and permanent identifiers. Social profiles may add employment context, but they can become outdated or may not distinguish between people with similar names.
Avoid copying claims about age, birthplace, family, salary, nationality, or private life unless a reliable first-party source has made that information relevant and public. The strongest available evidence concerns education, research, awards, publications, and professional specialization.
What Makes Jiajing Guan’s Profile Distinctive?
The most distinctive feature of Jiajing Guan’s public research profile is the combination of mathematical analysis and neural-network design.
Many machine-learning studies report a new architecture and compare benchmark errors. Her documented work places greater emphasis on why PINNs encounter difficulty in singularly perturbed problems and how tools from numerical analysis can reshape the learning problem.
Three themes stand out:
- Failure analysis: Identifying regimes where reconstruction or approximation becomes unreliable
- Structure-aware learning: Using equations, coordinate transformations, causality, and residuals
- Bridging methods: Combining classical solvers, reduced models, and neural networks instead of treating them as competing camps
That bridge is likely to remain important. Scientific computing has decades of theory and practical knowledge. Machine learning adds flexible approximation and optimization tools. Strong research asks how the two can reinforce each other without discarding mathematical safeguards.
Conclusion
Jiajing Guan is best understood through her work as an applied mathematician focused on difficult computational problems. Her documented path runs from undergraduate research in numerical mathematics and dynamical systems to a 2025 PhD centered on improving physics-informed neural networks for PDEs with steep gradients.
The practical lesson from her research is clear: scientific machine learning becomes more credible when it respects the structure of the equations, studies failure cases, and uses insights from established numerical methods.
Readers researching this name should rely on university records, the dissertation repository, and original publications. Those sources provide a far more accurate picture than short biography pages or automated profile databases.
Frequently Asked Questions
1. Who is Jiajing Guan?
Jiajing Guan is an applied mathematician whose public academic record includes research on numerical PDEs, dynamical networks, scientific machine learning, and physics-informed neural networks. She completed a PhD at the University of Maryland in 2025.
2. What did Jiajing Guan study?
Her doctoral research focused on improving PINNs for equations with steep gradients, especially Richards’ equation and convection-diffusion equations. The work explored input transformations, causality-aware residuals, adaptive architectures, and successive training.
3. Where did Jiajing Guan earn her PhD?
She earned her PhD through the University of Maryland’s Applied Mathematics, Statistics, and Scientific Computation program. Her advisor was Howard Elman.
4. What are transformed physics-informed neural networks?
Transformed PINNs alter the input coordinates so a neural network can devote more effective capacity to regions where a PDE solution changes sharply. Her 2024 paper examined this approach for convection-diffusion equations and analyzed the effect through neural tangent kernels.
5. Is Jiajing Guan a quantitative researcher?
A publicly indexed professional profile describes her as a quantitative researcher with a PhD in applied mathematics. Since job titles and employers can change, readers should verify the latest information through a current first-party professional profile.
