How Physics-Informed AI Can Build Better Climate and Engineering Models
0.02%. That’s the maximum energy drift observed in a simulated atmospheric model when researchers used a new physics-aware machine learning technique — compared to a 15% loss of energy over the same period using conventional data-driven modeling. This isn’t just a rounding error; it’s the difference between a climate simulator that stays physically plausible for decades and one that veers into unphysical territory within hours.
The culprit? Most AI-driven modeling ignores the bedrock laws of physics — conservation of energy, dissipation, stability — unless explicitly told not to. And when models are built from data alone, especially from black-box commercial software where the underlying equations are hidden, those laws often get broken.
But in a 2026 paper titled Petrov–Galerkin operator inference with application to stability-encouraging identification, Johannes Rettberg and colleagues introduce a method that changes this. They’ve developed a way to build fast, simplified models — called reduced-order models — directly from simulation data, while preserving core physical properties like stability and energy dissipation. Their approach, Petrov–Galerkin operator inference (PG-OpInf), doesn’t require access to the original code or equations. Instead, it uses a mathematically sophisticated projection framework rooted in port-Hamiltonian (pH) systems — a formalism that encodes energy flow and dissipation directly into the model structure.
This isn’t just theoretical elegance. For engineers designing micro-electromechanical systems, climate scientists running century-scale simulations, or biomedical researchers modeling tissue mechanics, the implications are profound: faster simulations that don’t break the laws of thermodynamics.
The Science
At the heart of this work is a tension between two worlds: the need for speed and the demand for accuracy. Full-scale simulations of complex systems — say, airflow over a wing or fluid movement in porous rock — can take days or weeks on supercomputers. To make them tractable, engineers use model order reduction (MOR): distilling a million-variable system into a few hundred that capture its essential behavior.
Traditional MOR methods are intrusive: they require direct access to the system’s governing equations. You project the full model onto a lower-dimensional space using techniques like Galerkin or Petrov–Galerkin projection. The latter, which uses different trial and test spaces, is particularly powerful because it can preserve stability and passivity — ensuring that systems don’t generate energy out of nowhere or oscillate uncontrollably.
But what if you can’t access the equations? Many industrial simulators are closed-source. You get input-output data, snapshots of states, maybe derivatives — but not the matrices that define the dynamics. That’s where nonintrusive methods come in, and among them, operator inference (OpInf) has gained traction.
Standard OpInf works like this: you collect time-series data — states $\bm{x}(t)$, their derivatives $\dot{\bm{x}}(t)$, inputs $\bm{u}(t)$, outputs $\bm{y}(t)$ — and assume the system follows a linear model:
Then, you project everything into a low-dimensional subspace spanned by a basis $\bm{V}$, typically obtained via proper orthogonal decomposition (POD). The reduced state is $\bm{x}_r(t) = \bm{V}^\intercal \bm{x}(t)$, and you solve a least-squares problem to find the best-fit reduced operators $\bm{A}_r, \bm{B}_r$ such that:
This is efficient and works well in many cases. But it uses a Galerkin projection — the trial and test spaces are the same ($\bm{W} = \bm{V}$) — which offers no guarantee of preserving stability or energy dissipation.
Rettberg et al. ask: what if we instead use a Petrov–Galerkin projection, where the test space $\bm{W}$ is chosen not arbitrarily, but based on physics?
Their key insight is that for port-Hamiltonian systems — which describe a vast range of physical phenomena from electrical circuits to fluid dynamics — there’s a natural choice: set $\bm{W} = \bm{Q}\bm{V}$, where $\bm{Q}$ is the Hessian of the Hamiltonian (essentially, the energy metric). This choice, known from intrusive MOR, ensures the reduced model remains passive and stable.
But $\bm{Q}$ is usually unknown in nonintrusive settings. So the authors propose two paths:
- PG-OpInf: Use known or estimated $\bm{Q}$ to define $\bm{W} = \bm{Q}\bm{V}$, then apply Petrov–Galerkin projection to infer operators.
- PG-OpInf-$\mathcal{H}$: If only scalar energy measurements $\mathcal{H}(t) = \frac{1}{2}\bm{x}(t)^\intercal\bm{Q}\bm{x}(t)$ are available, estimate $\bm{Q}$ from these.
- pHOpInf-CVX: Skip estimating $\bm{Q}$ entirely. Instead, formulate a convex optimization problem that directly infers a port-Hamiltonian structured model from data — no prior knowledge needed.
The last method is particularly striking: it’s a one-step, globally solvable convex program that enforces the entire pH structure — skew-symmetry of $\bm{J}$, positive semidefiniteness of $\bm{R}$ — without nonlinear optimization or fragile initialization.
What They Found
The authors tested their methods on four benchmark problems:
- A CD player model (mechanical vibrations)
- An atmospheric model (fluid dynamics with dissipation)
- A mass-spring-damper system (classical mechanics)
- A poroelasticity system (fluid flow in deformable porous media)
In every case, PG-OpInf significantly outperformed standard OpInf in both accuracy and physical plausibility.
Take the atmospheric model.
Energy Drift Over Time in Atmospheric Model Simulation
Comparison of energy conservation between standard operator inference and Petrov-Galerkin OpInf in an atmospheric model simulation. Standard method shows significant energy drift, while PG-OpInf maintains near-perfect energy preservation.
| Label | Value |
|---|---|
| t = 0 | 0 |
| t = 50 | 0 |
| t = 100 | 0 |
| t = 150 | 0 |
| t = 200 | 0 |
shows energy over time. The true system dissipates energy slowly due to viscosity. Standard OpInf loses 15% of energy within 200 time units — a catastrophic failure for long-term climate prediction. PG-OpInf, by contrast, stays within 0.02% of the correct energy level. It doesn’t just fit the data; it respects the physics.
Relative Operator Error Across Benchmark Problems
Relative Frobenius norm error in inferred system matrices compared to intrusive Petrov-Galerkin projection baseline.
| Label | Value |
|---|---|
| Standard OpInf | 0 |
| PG-OpInf | 0 |
compares operator errors — the Frobenius norm difference between inferred reduced operators and those from intrusive Petrov–Galerkin projection (the gold standard). On the poroelasticity system, standard OpInf had a relative error of 0.31; PG-OpInf reduced it to 0.05. That’s an 84% improvement in capturing the true reduced dynamics.
For the mass-spring-damper system, the authors also tested robustness to noise. Even with 10% additive noise in state derivatives, PG-OpInf maintained stable trajectories, while standard OpInf produced oscillations that grew over time — a hallmark of instability.
And in the convex pH identification (pHOpInf-CVX), the method successfully recovered physically meaningful $\bm{J}, \bm{R}, \bm{Q}$ matrices without any prior structural assumptions. The optimization converged reliably, avoiding the local minima that plague existing nonlinear approaches.
One subtle but critical finding was in the error analysis. The authors prove in Theorem 3.3 that as the residual (unresolved state components) shrinks, the nonintrusive operators converge to the intrusive ones — but only if the least-squares problem remains well-conditioned. In practice, this means there’s a sweet spot: too little data resolution and the model is inaccurate; too much, and numerical ill-conditioning corrupts the solution. This trade-off had been observed empirically but never formally bounded — until now.
Why This Changes Things
Most scientific machine learning focuses on fitting data better. This paper shifts the focus to fitting physics better.
Consider climate modeling. Global circulation models are among the most complex simulations ever built, requiring exascale computing. Reduced-order versions are essential for uncertainty quantification, ensemble forecasting, and real-time decision support. But if those reduced models violate energy conservation, their long-term predictions are suspect. A 15% energy drift isn’t just wrong — it invalidates the simulation.
PG-OpInf offers a path to fast, trustworthy surrogates. By building in energy-aware projections, it ensures that even simplified models respect the second law of thermodynamics. That’s not a minor detail; it’s what separates a useful emulator from a mathematical hallucination.
The same applies in engineering. The CD player benchmark may sound quaint, but it represents a class of flexible mechanical systems found in aerospace, robotics, and micro-devices. Stability isn’t optional: an unstable controller could destroy hardware. Traditional OpInf might learn a model that fits training data but fails under slight perturbations. PG-OpInf, by encouraging stability through its projection choice, builds in robustness from the start.
Perhaps most transformative is the convex pH identification method. Existing approaches to identifying port-Hamiltonian systems involve nonconvex optimization, heuristic initialization, and no guarantees of convergence. By reframing the problem as a convex program, the authors eliminate these pitfalls. As they write: “Unlike existing methods, we do not require knowledge of the underlying Hamiltonian, need to solve a nonlinear optimization problem, or rely on a process of fixing one matrix and optimizing the other.”
This opens the door to automated, reliable system identification in domains where safety and interpretability matter — power grids, medical devices, autonomous vehicles.
It also aligns with a growing movement in AI: moving from pure pattern recognition to structured learning, where models are constrained by domain knowledge. In climate science, this could mean embedding fluid dynamics symmetries; in biology, metabolic balance laws. PG-OpInf provides a template: identify the right projection — the right test space — and let physics guide the learning.
What's Next
The method has limits. Right now, it’s formulated for linear time-invariant (LTI) systems. Real-world systems are often nonlinear, time-varying, or stochastic. Extensions to nonlinear port-Hamiltonian systems — perhaps using lifting transformations or neural embeddings — are a natural next step.
Another challenge is data requirements. While PG-OpInf doesn’t need full access to operators, it does benefit from accurate state derivative estimates. In experiments, these must be approximated from noisy state measurements, which can amplify errors. Future work could integrate smoothing or differentiable programming to improve derivative estimation.
There’s also the question of scalability. The convex formulation is elegant, but solving large-scale semidefinite programs can be slow. Efficient algorithms tailored to the pH structure — exploiting sparsity, symmetry, or low-rank approximations — will be needed for high-dimensional systems.
Finally, the method assumes the full system is already port-Hamiltonian. But what if it’s not? Can we detect when a system should be pH-structured and still apply these tools? Or develop hybrid models that combine learned deviations with a physics-preserving core?
Still, the core idea — that the test space in model reduction should reflect physical principles — is likely to outlive this particular implementation. It suggests a broader design principle for scientific AI: don’t just project data into a low-dimensional space. Project it into a physically meaningful space.
As simulations grow more complex and data more abundant, the risk of building fast but untrustworthy models increases. Methods like PG-OpInf offer a counterbalance: speed without sacrificing truth. In a world where AI is increasingly used to model climate futures, design life-saving devices, and manage critical infrastructure, that balance isn’t just desirable — it’s essential.