Abstract
Physics-informed machine learning for the subsurface is usually met as a pile of acronyms rather than as a field with a shape. A practitioner reads about physics-informed neural networks, then about neural operators, then about reduced-order models and surrogates and hybrid schemes, and comes away with a list rather than a map. This survey reads the published work as one ordered family, separated by a single question: what does the method do with the governing physics? Five answers cover the ground. A PDE-constraint scheme folds the governing equation into the training loss, so the physics is a hard prior the fit must respect [1]. A reduced-order model projects an expensive full-order solve onto a small basis and keeps the physics upstream [5]. A neural operator or DNN surrogate learns the solution map from simulator runs and then answers downstream queries at a fraction of the cost [3] [4]. A hybrid physics-plus-data method keeps a physical prior and learns only the residual [2]. A pure data-driven net keeps no physics at all and fits the input-output relation straight from examples. Laid out on one spectrum, from physics fidelity on the left to data efficiency on the right, the families stop competing and start slotting into the regime each is good for, and the reported inference-speed payoff turns out to be ordered too: it concentrates toward the surrogate and data-driven end. From the period upstream survey we quote a reservoir-engineering DNN surrogate at 200x to 2,000x over a conventional simulator, a gradient-boosted production forecast at 100x or more, a geological-assessment DNN claiming up to a millionfold over manual mapping, and roughly 20 percent drilling cost savings [6]. This is a reading of the public field, not a reprise of our VeerNet architecture paper, and those figures are reported numbers we cite, not our own benchmark.
Background and related work
The organising idea the survey rests on is not ours; the field named it. Karniadakis and colleagues, reviewing physics-informed machine learning, describe a continuum that runs from strong physical priors at one end to purely data-driven models at the other, and argue that the right position on it is set by how much reliable physics and how much data a problem actually holds [2]. That sentence is the spine of this survey. Everything below is an attempt to place the concrete subsurface methods on that continuum honestly and to read the reported payoff along it.
The physics-heavy end has a precise anchor. Raissi, Perdikaris, and Karniadakis introduced physics-informed neural networks, which train a network to satisfy both the observed data and the governing partial differential equation by adding the equation's residual to the loss as a soft penalty [1]. The network is not merely fit to points; it is pushed toward solutions that obey mass balance, or diffusion, or whatever the governing relation is, at every collocation point sampled in the domain. For subsurface flow, where the physics of Darcy transport is written down and trusted, that is an attractive property: the model interpolates between sparse observations along a path the physics permits rather than one it merely tolerates.
The idea of trading an expensive solve for a cheap stand-in is older than the neural version and worth crediting on its own terms. Benner, Gugercin, and Willcox survey projection-based model reduction, where a high-dimensional full-order model is projected onto a low-dimensional subspace that captures the behaviour that matters, so a study needing thousands of runs pays for the cheap reduced model instead of the full one each time [5]. The reduced-order model keeps the physics; it just represents it in far fewer coordinates. The neural surrogate is that same bargain with a more flexible function class, and reading the two lineages next to each other keeps the survey honest about what is genuinely new and what is a rediscovery.
Operator learning sharpened the surrogate idea in the period just before this survey's close. Lu and colleagues introduced DeepONet, which learns a mapping between function spaces rather than a single input-output pair, so one trained network approximates the solution operator of a whole family of problems [3]. Li and colleagues introduced the Fourier neural operator, which parameterises that solution operator in the frequency domain and reports large speedups over a conventional solver at fixed accuracy, while generalising across resolutions [4]. Both are surrogates in the sense that matters here: the expensive physics runs upstream to generate training pairs, and the served model answers queries cheaply. They differ from a plain regression net in learning an operator, not a point, which is why a single trained model can stand in for a parametric sweep.
The reported payoff that this survey reads along the spectrum comes from one period upstream survey. Koroteev and Tekic aggregate the headline acceleration and cost figures for machine learning across upstream oil and gas: a reservoir-engineering DNN surrogate reported at 200x to 2,000x over a conventional simulator, a production-optimisation model based on gradient boosting reported at 100x or more for well forecasting, a geological-assessment DNN reported at up to a millionfold over manual mapping, and roughly 20 percent cost savings from ML-based drilling optimisation [6]. We treat those as reported figures with named baselines, not as universal constants, and the survey's contribution is to show that they are not scattered: they line up with position on the physics-to-data spectrum.
Method
This is a structured reading of the published physics-informed machine-learning literature as it stood at the close of the survey's own quarter, not a new experiment. The procedure was kept narrow so the claims stay defensible. We began from the spectrum the field's own review names, physics fidelity at one pole and data efficiency at the other [2], and asked of each method the single sorting question: what does it do with the governing physics? That question partitions the concrete subsurface methods into five families, and we placed each family on the spectrum by how much governing physics it retains: PDE-constraint schemes that put the equation in the loss [1], reduced-order models that project the full-order physics onto a small basis [5], neural operators and DNN surrogates that learn the solution map from runs of the physics [3] [4], hybrid schemes that keep a prior and learn a residual [2], and pure data-driven nets that keep no prior at all.
For each family we recorded three things: what it does with the physics, what it costs to build and to serve, and the regime its authors claim it for. We then overlaid the reported acceleration figures from the period upstream survey onto the same spectrum, attaching each figure to the family it describes: the reservoir surrogate band and the production-forecast speedup to the surrogate family, the mapping speedup to the data-driven end, and the drilling cost saving as a cost rather than a speed figure [6]. The interactive exhibit below is built on that footing. Its horizontal axis is the physics-to-data spectrum; its vertical axis carries the sourced reported speedup on a log scale, with a lever to switch it to an illustrative physics-fidelity ranking so the trade reads directly. Only the speedup figures and the drilling saving are sourced; the physics-fidelity ordinal is a flagged illustrative ranking, and the map orders the families rather than benchmarking them head to head.
To keep the survey anchored to a real task rather than to abstractions, we read it against one point at the data-driven pole drawn from our own work: VeerNet, a raster well-log digitiser that reconstructs depth-indexed curve values from scanned images with no flow equation anywhere in it. It is a pure data-driven net by the survey's own definition, and it is here only to give the far-right end of the spectrum a concrete example, not to benchmark it against the physics-heavy families it does not compete with.
The spectrum, read left to right
The survey's whole claim is that these methods form an order, so it is worth walking the spectrum once, pole to pole, before reading the payoff along it.
At the physics-heavy pole sits the PDE-constraint scheme. A physics-informed neural network encodes the governing equation as a residual added to the loss, so the trained model is pulled toward physically admissible solutions rather than merely plausible ones [1]. Its strength is exactly where data is sparse and physics is trusted: it can honour a governing relation between the few observations it has. Its cost is that the physics is paid on every training step, and, as the period's own follow-up work documents, balancing the PDE-residual term against the data term is delicate enough to be a research topic of its own [7].
One step toward data sits the reduced-order model. It keeps the full physics but represents it in a small basis, so the served model is cheap while the physics remains auditable upstream [5]. It is the least fashionable family in the current literature and often the most defensible one when the governing model exists and is trusted, because the reduction is a controlled approximation rather than a learned guess.
In the middle sits the neural operator and DNN surrogate, and this is the family that carries most of the reported speed. It runs the expensive physics enough times to generate training pairs, fits a network to imitate the solution map, and then answers thousands of downstream queries against the network at a fraction of the cost [3] [4]. The physics is still in the loop, upstream, where it trained the surrogate; what changed is that it is no longer paid on every query. This is where the reservoir-surrogate band of 200x to 2,000x over a conventional simulator attaches [6].
Further toward data sits the hybrid physics-plus-data method, which keeps a physical prior and learns only the residual the prior gets wrong [2]. It is the pragmatic middle: use the physics you trust, learn the part you cannot write down. The production-optimisation speedup of 100x or more, from a gradient-boosted forecaster standing next to rather than inside a reservoir model, reads most naturally here [6].
At the data-heavy pole sits the pure data-driven net, which keeps no physics at all and fits the mapping straight from examples. This is the right trade when the governing relation is unknown or is not the point. Reconstructing a curve trace off a scanned log is a perception problem, not a reservoir-physics one, which is why VeerNet lives here honestly rather than as a concession, and it is also where the reported millionfold mapping speedup sits, against a manual-labour baseline rather than a simulator one [6].
Reading the payoff along the spectrum
The exhibit makes the survey's second claim legible: the reported inference-speed payoff is not scattered across the families, it climbs toward the data-driven end. The physics-heavy pole reports no headline speedup in the period survey at all, because a PDE-constraint scheme is not primarily a speed play; it is an accuracy-and-admissibility play for the sparse-data, trusted-physics regime [1]. The surrogate family carries the first large, honestly-comparable numbers, the 200x to 2,000x reservoir band, because that is exactly the case where a cheap learned stand-in replaces a genuinely expensive full-order solve query by query [6]. The data-driven pole carries the largest number, the millionfold mapping figure, precisely because its baseline is the slowest thing in the comparison, a human doing the mapping by hand, which is the point the honest reader must hold onto: a speedup is a ratio, and the denominator is doing a lot of the work [6].
That last observation is why the map orders rather than ranks. It would be a category error to line up 2,000x against 1,000,000x and conclude the data-driven net is five hundred times better than the surrogate. The two numbers have different denominators, a conventional simulator versus a person, and different tasks, a reservoir solve versus a geological map. What the ordering does show, and all it claims to show, is that the reported speed payoff systematically favours methods that pay less for physics at serve time, and that the drilling cost saving of roughly 20 percent sits off the speed axis entirely as a cost figure, not an acceleration [6]. Position on the physics-to-data spectrum predicts the shape of the payoff, not a single scalar of merit.
Discussion
Read as one family, the five methods answer a placement question rather than a partisan one. If the governing physics is known, trusted, and data is sparse, the PDE-constraint scheme is the natural pick, and its cost is training difficulty rather than serving cost [1] [7]. If the physics is trusted but a study needs thousands of runs, the reduced-order model or the neural surrogate is the move, and the surrogate is where the reported reservoir speedups live [4] [5] [6]. If some physics is trusted and some is missing, the hybrid keeps the prior and learns the residual [2]. And if the governing relation is unknown or beside the point, the pure data-driven net is the honest choice, not a compromise. The spectrum the field's own review names is the tool that turns a pile of acronyms into that placement decision [2].
Where our own work sits relative to this literature is worth marking, because it is the line between this survey and our applied writing. This is a reading of how the public field arranges its physics-informed methods and where the reported payoff falls along that arrangement. VeerNet is one concrete pin at the data-driven pole, a digitiser with no physics prior solving a perception task, and it is downstream of the survey, not its subject. The survey explains why a subsurface team should read these methods as positions on a spectrum rather than as camps to join, and why the reported speed figures, striking as the millionfold headline is, are ratios whose baselines have to be named before they mean anything.
Limitations
This is a survey and inherits a survey's limits. It synthesises what the published physics-informed machine-learning and upstream-AI literature reports and does not re-implement or re-measure any of the methods it discusses. The acceleration figures it reads along the spectrum, the 200x to 2,000x reservoir band, the 100x-or-more production forecast, the up-to-a-millionfold mapping speedup, and the roughly 20 percent drilling cost saving, are reported numbers from one period upstream survey with named but heterogeneous baselines, a conventional simulator in one case and a human mapper in another, and they must not be compared as if they shared a denominator [6]. The survey therefore orders the families and does not rank them; it makes no claim that any one family is universally faster or better, only that the reported payoff correlates with how little physics a method pays for at serve time. The physics-fidelity axis in the exhibit is an illustrative ordinal ranking of how much governing physics each family retains, flagged as such on the canvas, and is not a measured quantity. The single point from our own work, VeerNet at the data-driven pole, is a concrete example of the far-right end and not a benchmark against the physics-heavy families it does not compete with. The survey scopes itself to five families and to the literature as it stood at the close of its own quarter, so later refinements of neural operators, physics-informed training, and hybrid schemes that the field has since explored are out of frame. A reader should take this as a map of where a given subsurface method sits and what kind of payoff to expect from that position, not as a substitute for measuring a chosen method against the specific baseline their own problem defines.
What to carry from the survey
- Physics-informed ML for the subsurface is one ordered family, not a pile of acronyms. The sorting question is what a method does with the governing physics, and the answers place it on a single spectrum from physics fidelity to data efficiency.
- Five families cover the ground: PDE-constraint schemes that put the equation in the loss (physics-informed neural networks), reduced-order models that project the full-order solve onto a small basis, neural operators and DNN surrogates that learn the solution map from simulator runs, hybrid physics-plus-data methods that keep a prior and learn the residual, and pure data-driven nets that keep no physics at all.
- The reported inference-speed payoff climbs toward the data-driven end. The physics-heavy pole reports no headline speedup because it is an accuracy play; the surrogate family carries the 200x to 2,000x reservoir band; the data-driven pole carries the up-to-a-millionfold mapping figure (Koroteev and Tekic 2021).
- A speedup is a ratio, and the denominators differ. The 2,000x surrogate figure is measured against a conventional simulator; the millionfold mapping figure is measured against a human doing the mapping by hand. The map orders the families rather than ranking them, and the roughly 20 percent drilling saving sits off the speed axis as a cost figure.
- Read this way, choosing a method is a placement decision: trusted physics and sparse data favour a PDE-constraint scheme; trusted physics and many runs favour a reduced-order model or surrogate; partial physics favours a hybrid; an unknown or irrelevant governing relation favours a pure data-driven net, which is where a raster-log digitiser like VeerNet honestly lives.
The smallest habit this survey would install is a question to ask before picking a method at all: how much of the governing physics do I actually trust here, and how much data do I actually have, because the answer places the problem on the spectrum, and the spectrum, not fashion, is what should choose between a PDE-constraint scheme, a surrogate, and a data-driven net.
References
[1] Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics 378 (2019), pp. 686-707. The PDE-constraint pole: the governing equation enters the loss as a soft residual penalty. https://www.sciencedirect.com/science/article/pii/S0021999118307125
[2] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., and Yang, L. Physics-informed machine learning. Nature Reviews Physics 3 (2021), pp. 422-440. The review that names the physics-to-data spectrum this survey is organised around. https://www.nature.com/articles/s42254-021-00314-5
[3] Lu, L., Jin, P., Pang, G., Zhang, Z., and Karniadakis, G. E. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature Machine Intelligence 3 (2021), pp. 218-229. Learns a mapping between function spaces, the operator-learning form of a neural surrogate. https://www.nature.com/articles/s42256-021-00302-5
[4] Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A., and Anandkumar, A. Fourier Neural Operator for Parametric Partial Differential Equations. ICLR (2021). Parameterises the solution operator in the frequency domain and reports large speedups over a conventional solver at fixed accuracy. https://arxiv.org/abs/2010.08895
[5] Benner, P., Gugercin, S., and Willcox, K. A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Review 57(4) (2015), pp. 483-531. The reduced-order-model lineage that predates the neural surrogate and frames the same replace-the-expensive-solve idea. https://epubs.siam.org/doi/10.1137/130932715
[6] Koroteev, D., and Tekic, Z. Artificial intelligence in oil and gas upstream: Trends, challenges, and scenarios for the future. Energy and AI 3 (2021), 100041. The source of the reported acceleration figures read along the spectrum: reservoir surrogates 200x to 2,000x over a conventional simulator, production optimisation 100x or more via gradient boosting, geological assessment up to a millionfold over manual mapping, and about 20 percent drilling cost savings. https://www.sciencedirect.com/science/article/pii/S2666546820300410
[7] Wang, S., Teng, Y., and Perdikaris, P. Understanding and mitigating gradient flow pathologies in physics-informed neural networks. SIAM Journal on Scientific Computing 43(5) (2021), pp. A3055-A3081. Diagnoses why balancing the PDE-residual loss against the data loss is hard, the practical cost of the PDE-constraint pole. https://epubs.siam.org/doi/10.1137/20M1318043
