Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves

arXiv:2610.09510 · cs.LG, cs.NA, math.NA · Submitted 2026-10-07 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Physics-Informed Neural Plasticity".

Jane: The gist Physics-informed neural plasticity introduces Representation Capacity Adaptation for PDEs (ReCAP), a paradigm where the representation itself reshapes during optimization in response to unresolved physics,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we’re looking at this paper, "Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves." It sounds like a major shift because it tackles the problem that current physics-informed neural networks have—that they keep the same structure throughout training, which isn't good for complex problems.

Jane: Right. So instead of having one fixed set of neurons or features to solve everything, this approach suggests the representation itself should change while we're optimizing. It’s like being able to dynamically adjust your toolbox as you see what tools you actually need for a specific job, which is what they call physics-informed neural plasticity.

Lu: What’s interesting about this paper is how they realize that by making the representation plastic, the solver can adapt where it needs more capacity and where it doesn't need it. They introduce something called ReCAP, which is this Gaussian-localized solver that reshapes capacity through a few specific mechanisms.

Meng: Reshaping capacity sounds ambitious for production. How do you actually manage that change without the training process becoming totally unstable? What are the specific knobs they turn to control this redistribution?

Lalam: The paper explains ReCAP operates by alternating between optimizing the continuous parameters of the representation and then updating that representation using localized physics errors, which is how it manages these structural changes.

Tom: Okay, so let's talk about what this ReCAP actually does mechanically. They have four primary operations for modification: polynomial enrichment, gate-based pruning, functional merging, and residual-guided Gaussian splitting. These are the ways it redistributes that capacity we talked about earlier.

Jane: Those sound like very specific actions on the representation geometry. Can you unpack what polynomial enrichment means in plain terms for someone who isn't deep in the math?

Lu: Polynomial enrichment just means if a component is making too much error in a certain spot, you increase its polynomial order without adding any new components to the model, as long as that local error indicator eta m is above a threshold called tau split eta e.

Meng: So it's like boosting the expressiveness of an existing feature instead of just throwing in a whole new neuron or Gaussian component. That sounds like a way to use existing resources more efficiently.

Tom: Exactly, and then you have gate-based pruning where they look at components with small gate probabilities, gamma m is the sigmoid of g m, and they prune those down. It’s trimming the fat out of the representation.

Title and authors: Jane: And then there's functional merging, which sounds like they’re looking to simplify things by combining redundant parts. What makes them decide if two components should merge together?

Lu: They accept a pair only if they have both non-negligible effective overlap, which is measured by O ij being greater than tau overlap, and their local predictions are similar, meaning D ij is less than tau sim.

Meng: So it’s not just about them having some kind of shared connection; they have to be actually predicting the same thing locally. That sounds like a useful filter for reducing redundancy during training.

Tom: And finally, residual-guided Gaussian splitting, which is how they introduce brand new capacity where the physics is unresolved. They choose the split direction based on the residual-weighted responsibility, which means they pick an axis determined by the principal eigenvector of the residual-covariance matrix C.

Jane: That sounds a lot more intelligent than just throwing random new features into a gap. So, to summarize how ReCAP works—it uses these four operations guided by an adaptation score eta m = eta res m + lambda bdry eta bdry m + lambda jump eta jump m to decide what to do next.

Lu: That adaptation score couples the local physics enrichment, the boundary condition errors, and the jump errors together, giving them a single ranking for structural change decisions.

Meng: If they are using that score to decide if they should enrich a component or split capacity in an unresolved region, what happens when they try to deploy this on a massive system? Does it get computationally prohibitive?

Tom: The computational cost analysis shows ReCAP uses zero point five three GFLOPs on KGthree dee, which is matching SINN and falling below ActNet at one point six four, PirateNets at two point five nine, and CoupledNet at four point three eight. It’s quite efficient for what it achieves in terms of error reduction on the benchmarks tested.

Jane: The experimental validation is really strong here too, because they tested ReCAP across five three dee and 4D PDE benchmarks against eleven competing solvers, and ReCAP achieved the lowest relative L two error on every single one of them <ref:2610.09510#pg3,the lowest relative $L_2$ error on every>.

Lu: Specifically, on FMthree dee they reduced the error by twenty-seven point five percent compared to the strongest baseline result, and on Hthree dee they cut it by ten point seven percent. The qualitative visualizations also showed that ReCAP resolves the dominant solution structures while keeping errors small and spatially localized.

Meng: If I were deploying this, I’d be very concerned about those guarantees they have for stability. Do we know how reliable these structural changes are under real-world, noisy data conditions?

Title and authors: Tom: They establish conditional a posteriori reliability and structural-stability guarantees linking the localized physics residuals to the solution error and stable refinement. Theorem B.one gives a mass-weighted ReCAP reliability certificate, showing that the global residual norm is reconstructed by mass weighting local indicators, which bounds the perturbation from additive refinement.

Jane: So it’s not just about getting low errors on a test set; they are providing a mathematical way to trust that the solver is actually doing what it’s supposed to do according to the underlying physics. That kind of certification is really important for adoption.

Lu: They also have Theorem B.four which proves that the default additive quiet-child split produces a controlled instantaneous perturbation under specific assumptions about how capacity grows with controlled disruption <ref:2610.09510#pg3>.

Meng: That controlled perturbation part is what I care about when I think about scaling up. If the capacity growth is predictable and bounded, then it’s much more deployable than something where it just explodes unpredictably.

Tom: So, to wrap this up for the mechanics, ReCAP treats representational capacity as an adaptive resource for solving physics-informed PDEs by jointly learning the solution and adapting where, how much capacity is used to represent it.

Jane: It’s a sophisticated way to handle problems where the complexity across space and time isn't uniform, allowing the solver to be flexible instead of rigid. What this means for us is that we can build solvers that are inherently more efficient when dealing with those hard, heterogeneous problems in science and engineering.

Lu: It suggests a future where representation isn't just a fixed mapping but an active part of the optimization process, which is what physics-informed neural plasticity aims to achieve.

Meng: For deployment, it means we might need to design our training pipeline not just around the final model, but around how we allow its internal structure to evolve during learning. That’s a significant shift in how we think about model architecture development.

Lalam: I’m confident in the results because they show consistency across different problem types and they provide those formal guarantees on error reconstruction. It's a very solid contribution to the field of PINNs.

Tom: Well, that’s our take on Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves. We’ll be looking at these results and see how this adaptive representation idea fits into the broader landscape of generative models next time.

The paper's summary: Tom: So we're looking at this paper, "Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves." Essentially, it’s about taking physics-informed neural networks and making them plastic so the representation—the internal structure of the AI—actually changes while it's solving a problem.

Jane: That sounds like the idea that instead of having one rigid set of tools for every job, you have a toolbox that grows and shrinks based on what kind of job you're doing right now. It’s about letting the representation itself reshape in response to physics errors you haven't fully resolved yet.

Lu: They realize standard PINNs have a fixed structure, which isn't good when the complexity across space and time is super messy. So this paper introduces ReCAP, which is their method for this plastic representation adaptation.

Meng: I’m wondering how they manage that structural change without the whole training process just falling apart? Does it get computationally impossible to track all those changes?

Lalam: ReCAP works by alternating between optimizing the continuous parameters and then updating that representation using localized physics errors. They have these specific adaptation rules for handling those structural shifts.

Tom: Right, so they have four main ways the representation is modified: polynomial enrichment, gate-based pruning, functional merging, and residual-guided Gaussian splitting. That’s how they dynamically redistribute capacity where it's needed or where it's just redundant.

Jane: Polynomial enrichment sounds like boosting an existing feature instead of adding a whole new one if that specific spot is causing high error. And functional merging—that’s when they combine two parts only if they are actually predicting the same thing locally, which helps clean up redundancy.

Lu: They use an adaptation score, eta m, which combines local physics residuals with boundary condition errors and jump errors to decide what modification to apply next. It's a way of ranking candidates for change.

Meng: So they’re using this score to make decisions about the structure, which is cool, but what about deployment? If we try to run this on a massive simulation, does it become too slow just managing all those structural updates?

Tom: The cost analysis shows ReCAP uses zero point five three GFLOPs on KGthree dee. That's actually quite efficient when you compare it to some of the other solvers they tested, like ActNet or CoupledNet which are running around one point six four and four point three eight respectively.

Jane: And the experimental results are really telling here because they tested this against eleven different physics-informed solvers across five three dee and 4D benchmarks, and ReCAP consistently achieved the lowest relative L two error on every single one of them <ref:2610.09510#pg3>.

Lu: On FMthree dee, for instance, they reduced the error by twenty-seven point five percent compared to the strongest baseline result. They show that this adaptive approach is better at capturing the actual solution structures while keeping those errors small and localized in space.

Meng: So it's working well on these test cases, but I still need to know about reliability when we move beyond perfect benchmarks to real-world data where things are noisy. What guarantees do they actually have about the stability of this reshaping?

Tom: They establish conditional a posteriori reliability by linking those local physics indicators back to the solution error. Theorem B.one gives them a mass-weighted certificate showing that the global residual norm is reconstructed by weighting those local indicators, which bounds how much the perturbation from adding refinement can be.

Jane: That's a big deal for trust. It means they aren't just reporting low error numbers; they have a mathematical way to prove that the solver is actually following the physics constraints reliably under certain conditions.

Lu: Plus, Theorem B.four proves that their default splitting method—the quiet-child update—actually produces a controlled instantaneous perturbation when you follow specific assumptions about how capacity grows during disruption.

Meng: Controlled perturbation sounds much safer than unpredictable chaos when we're shipping models to production pipelines where stability is everything.

Tom: So, to sum up the core idea, ReCAP treats representational capacity as an adaptive resource for solving physics-informed PDEs by jointly learning the solution and adapting where, how much capacity is used to represent it.

Jane: It suggests a future where representation isn't just a fixed mapping but an active part of the optimization process, which is what this physics-informed neural plasticity aims to achieve.

Lu: This means we can build solvers that are inherently more flexible when dealing with those hard, heterogeneous problems in science and engineering.

Lalam: I'm confident in the results because they show consistency across different problem types and they provide those formal guarantees on error reconstruction. It's a very solid contribution to the field of PINNs.

The paper's improvements: Tom: So we're looking at how they are improving their ReCAP method, which is about making sure these structural changes are actually stable and controlled when they happen during training.

Jane: They’re adding a few specific knobs to manage that instability, focusing on making sure the new capacity introduced doesn't just create noise. It’s not just about adapting where things are; it’s about controlling the *how much* and *when* of the adaptation.

Lu: They introduced these four operations—enrichment, pruning, merging, and splitting—but they also tied them to a specific adaptation score eta m that couples physics residuals with boundary conditions. That coupling is key for controlling the process.

Meng: So if I’m trying to deploy this on a real system, I need assurance that these rules don't cause catastrophic failure because the capacity shifts too violently. What stops it from just blowing up?

Tom: They have theorems that provide conditional a posteriori reliability, which means they give you a mathematical way to check if the global physics loss is correctly reconstructed by those local indicators. It links the change back to known stability bounds on the PDE.

Jane: That sounds like they’re providing a safety net for researchers who are using this, showing them that their structural changes are tied directly to actual physical errors, not just random fluctuations in the AI training process.

Lu: They also have theorem B.four which deals with the quiet-child split, and it proves that under certain assumptions about capacity growth, this method produces a controlled instantaneous perturbation. That means when they introduce new capacity into an unresolved region, it does so predictably.

Meng: Predictable is good for production, but how much control do we really have over those assumptions? Are those assumptions easy to meet when the physics are genuinely complex?

Tom: The paper shows that the default additive quiet-child split works under specific conditions, meaning you get a predictable change in function if you stick to their rules about how capacity grows. It’s not magic; it’s a controlled procedure.

Jane: So what this changes for someone listening to the show is that instead of just seeing a solver that gets low error numbers on paper, they see one with built-in mechanisms to manage its own internal structure in a way that is mathematically traceable and constrained.

Lu: It moves the field toward solvers where you aren't just teaching the AI a mapping; you’re designing an AI whose internal architecture evolves dynamically based on what it’s struggling with in real time.

Meng: If this works, it means we can tackle those really hard problems—the ones where the physics is super heterogeneous—without having to hand-tune the entire model structure manually beforehand. That sounds like a huge reduction in manual effort for complex simulations.

Tom: It really shifts the focus from just training an AI to designing an adaptive system that manages its own complexity during optimization.

Jane: And that brings us right back around to how this idea of representation plasticity interacts with other areas we've been hearing about, like how models handle memorization in diffusion processes.

Conclusion: Tom: So to wrap up, this paper introduces Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves as a way to make the representation itself an adaptive resource during optimization.

Jane: It basically means we can build solvers that are flexible enough to grow or shrink their internal structure based on the physics they are trying to solve, which is pretty powerful for problems with really messy complexity.

Lu: The main implication is moving away from fixed neural network architectures towards systems where the representation learns and adapts where it needs more capacity, which should help with those hard PDE benchmarks we’ve seen.

Meng: So for me, if this works on a real simulation, it means less time spent on manual model design and more time focusing on the training data itself to guide that adaptation. That changes the deployment workflow significantly.

Lalam: I'm sure of my answer here; by providing those formal reliability guarantees, they've made this kind of structural plasticity much more trustworthy for serious scientific work than just a proof of concept.

Tom: Yeah, it’s solid because they gave us those numbers on efficiency, like zero point five three GFLOPs on KGthree dee, and showed they beat several established baselines in terms of error reduction.

Jane: It really shows that the more complex a problem gets across space and time, the more necessary this kind of dynamic adjustment becomes for an AI solver to actually succeed.

Lu: This opens up a lot of avenues for future work, especially connecting this idea of plastic representations to how models handle learned energy landscapes in diffusion processes.

Meng: I’m curious if we can see any practical applications beyond just solving these specific PDE benchmarks, maybe something related to optimizing complex engineering simulations.

Lalam: They’ve laid a good foundation for that, and I think the formal guarantees they provided on error reconstruction are what makes this paper stand out in the literature.

Tom: It’s definitely a big step in treating representation capacity as something that can actually be managed and optimized during the training process.

Jane: And it suggests we need to start thinking about our AI architectures not just as static tools, but as evolving systems that can respond to the underlying physics they are trying to model.

Lu: Exactly, we need those mechanisms for structural plasticity if we want generative models to handle things that are genuinely unpredictable in their complexity.

Meng: I'm still focused on making sure this doesn't require an impossibly complex setup just to manage these adaptation rules efficiently at scale.

Lalam: Overall, Physics-Informed Neural Plasticity: PDE Solvers That Reshape Themselves offers a very concrete path toward more robust and efficient solvers for high-dimensional problems.

Chun-Wun Cheng, Bingcheng Hu, Angelica I. Aviles-Rivero

University of Cambridge · Shanghai Normal University · Tsinghua University

cs.LG, cs.NA, math.NA

Submitted: 2026-10-07

Updated: 2026-10-07

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 84/100

The gist: The gist Physics-informed neural plasticity introduces Representation Capacity Adaptation for PDEs (ReCAP), a paradigm where the representation itself reshapes during optimization in response to

Key concepts

Physics-Informed Neural Plasticity: This paradigm means the neural network's structure (its representation) is not fixed. Instead, it actively reshapes itself during training in response to physics errors that remain unresolved. This allows the solver to adapt its internal structure dynamically based on the complexity of the problem.
ReCAP Mechanism: ReCAP is a Gaussian-localized solver that manages capacity redistribution through several adaptation rules. It alternates between optimizing continuous parameters and updating the representation using localized physics errors, guided by an adaptation score that combines residual error, boundary conditions, and jump indicators.
Polynomial Enrichment: This operation increases the local expressivity of a component without adding new components. It works by increasing the active polynomial order of high-error components if their current adaptation score exceeds a threshold, effectively refining the existing representation locally.
Residual-Directed Gaussian Splitting: This technique introduces new representation capacity in regions where physics is unresolved. The direction for this split is determined by selecting the principal eigenvector of the residual-covariance matrix, ensuring that new capacity is added along the direction of dominant residual energy spread.

Summary

The gist Physics-informed neural plasticity introduces Representation Capacity Adaptation for PDEs (ReCAP), a paradigm where the representation itself reshapes during optimization in response to unresolved physics, allowing solvers to adapt where, how, and how much capacity is used to represent it

Physics-Informed Neural Plasticity

The core idea is that physics-informed neural networks (PINNs) typically have a fixed representation structure, which is insufficient for solutions with strongly heterogeneous complexity across space and space–time. The authors introduce physics-informed neural plasticity, where the representation itself reshapes during optimization in response to unresolved physics. This principle is instantiated through ReCAP, a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging.

ReCAP Mechanism

ReCAP operates by alternating between optimizing the continuous parameters of the representation and updating the representation using localized physics errors. The adaptation rules act at different levels to manage structural changes. The adaptation scores are determined by a complete adaptation score ηm = ηres m + λbdryηbdry m + λjumpηjump m, which couples local p-enrichment with residual-directed h-like splitting.

Adaptation Operations

The paper details four primary operations for representation modification:

Polynomial enrichment: This increases local expressivity without adding components by increasing the active polynomial order of high-error components if ηm > τsplitηe.
Gate-based pruning: This targets components with small gate probabilities, where γm = sigmoid(gm).
Functional merging: This coarsen redundant components by accepting pairs only if they exhibit both non-negligible effective overlap (Oij > τoverlap) and similar local predictions (Dij < τsim).
Residual-guided Gaussian splitting: This introduces new capacity in unresolved regions by selecting a split direction based on the residual-weighted responsibility, where the split direction is chosen as the principal eigenvector of the residual-covariance matrix C.

Theoretical Guarantees

The authors establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Theorem B.1 provides a mass-weighted ReCAP reliability certificate, showing that the global residual norm is reconstructed by mass weighting local indicators, and it bounds the perturbation from additive refinement. Furthermore, Theorem B.4 proves that the default additive quiet-child split produces a controlled instantaneous perturbation under specific assumptions.

Experimental Validation

ReCAP was tested across five challenging 3D and 4D PDE benchmarks against eleven physics-informed solvers. Across these problems, ReCAP achieves the lowest relative L2 error on every benchmark, reducing error by 10.7%–27.5% relative to the strongest competing result. Qualitative visualizations confirm that ReCAP accurately resolves dominant solution structures while maintaining small and spatially localized errors.

Computational Efficiency

The computational cost analysis shows that ReCAP uses 0.53 GFLOPs on KG3D, matching SINN and falling below ActNet (1.64), PirateNets (2.59), and CoupledNet (4.38).

Conclusion

ReCAP treats representational capacity as an adaptive resource for physics-informed PDE solving, jointly learning the solution and adapting where, how, and how much capacity is used to represent it.

The gist The paper introduces physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics.

How it works

ReCAP realizes physics-informed neural plasticity by adapting the approximation space together with its trainable parameters, alternating between optimizing continuous parameters and updating the representation using localized physics errors. The adaptation rules act at different levels to manage structural changes, where the base objective Lbase determines how the PDE and its conditions are enforced, while adaptation scores determine which components are modified.

Representation and Local Error Indicators

The representation consists of Gaussian-localized components Gm = (µm,σm, gm, cm, pm), where ξm(z) is the normalized coordinate of component m. The component-wise residual indicator ηres m = PNc i=1 πm(zi)rΘ(zi)2PNc i=1 πm(zi) + ε indicates a large weighted PDE residual in the region for which component m is responsible. Additionally, ηbdry m measures the condition-error indicator using value observations. The complete adaptation score is ηm = ηres m + λbdryηbdry m + λjumpηjump m, which ranks candidates for modification.

Polynomial Enrichment, Pruning, and Functional Merging

The representation is modified through polynomial enrichment, pruning, merging, and Gaussian splitting based on the median error ηe = medianm ηm. Polynomial-order refinement increases local expressivity by activating preallocated polynomial terms if ηm > τsplitηe. Gate-based pruning targets components with small gate probabilities where γm = sigmoid(gm). Functional merging coarsen redundant components by accepting pairs only if they exhibit both non-negligible effective overlap (Oij > τoverlap) and similar local predictions (Dij < τsim).

Residual-Directed Splitting and Coordinate-Aware Initialization

Splitting introduces new capacity in unresolved regions, with the split direction chosen as the principal eigenvector of the residual-covariance matrix C, oriented along the dominant spread of residual energy. The child geometry is initialized using a quiet-child update where its polynomial coefficients are translated into shifted and rescaled coordinates to control instantaneous functional perturbation.

Conditional Reliability and Control

The authors establish conditional a posteriori reliability, showing that the mass-weighted local indicators reconstruct the global physics loss. Theorem B.1 proves that under stability assumptions, the solution error is bounded by a combination of mass-weighted local indicators and sample-count dependent terms.

Experimental Results

ReCAP achieves the lowest reported relative L2 error on each of the five benchmarks in Table 1. Compared with the strongest baseline on each problem, ReCAP reduces error by 27.5% on FM3D (PirateNets), 10.7% on H3D (SPINN), 12.1% on ND3D (PIG), 15.7% on KG3D (SPINN), and 12.2% on KG4D (JAXPI).

Computational Cost

ReCAP uses 0.53 GFLOPs in this comparison, matching SINN and falling below ActNet (1.64), PirateNets (2.59), and CoupledNet (4.38).

The gist The paper introduces physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics, allowing solvers to adapt where, how, and how much capacity is used to represent it.

Improvements for AI systems

Bold header: Adaptive Representation Capacity for PDEs (ReCAP) implementation. ReCAP can dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging, allowing it to allocate representational resources precisely where the physics is difficult or redundant.
Bold header: Controlled refinement via quiet-child update. The system can employ quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation, ensuring that structural changes are both targeted and stable, as supported by Theorem B.4's bound on the loss increase.
Bold header: Residual-guided splitting direction. Instead of relying on fixed geometric rules, ReCAP can use a more informative alternative where the split direction is chosen as the principal eigenvector of a residual-weighted covariance, meaning refinement is oriented along the direction in which the responsibility-weighted residual exhibits the largest spread.
Bold header: Function-aware merging criterion. The system can distinguish between merely overlapping components and functionally similar ones by accepting pairs only if they satisfy both effective overlap and agreement of local predictions, ensuring that merging only consolidates redundant representations where both exhibit non-negligible effective overlap and similar local functions.
Bold header: A posteriori reliability certification. ReCAP can provide a quantitative measure of trust by establishing conditional a posteriori reliability through Theorem B.1, which connects mass-weighted local physics indicators to solution error, allowing researchers to verify the solution accuracy against known stability bounds on the PDE.

Abstract

Physics-informed neural PDE solvers adapt their parameters to satisfy governing equations, yet their representational structure typically remains fixed throughout training. This rigidity is poorly matched to PDE solutions with strongly heterogeneous complexity across space and space--time, leaving capacity insufficient where the physics is difficult and redundant where it is simple. We introduce physics-informed neural plasticity, a paradigm in which the representation itself reshapes during optimization in response to unresolved physics. We instantiate this principle with Representation Capacity Adaptation for PDEs (ReCAP), a Gaussian-localized solver that dynamically redistributes capacity through local enrichment, residual-directed splitting, gate-based pruning, and function-aware merging. ReCAP uses responsibility-weighted error indicators and the geometry of residual energy to determine where and how to refine. To limit the disturbance introduced by splitting, we introduce quiet-child refinement, which initializes new components by transporting the parent representation while controlling instantaneous functional perturbation. We further establish conditional a posteriori reliability and structural-stability guarantees linking localized physics residuals to solution error and stable refinement. Across five challenging 3D and 4D PDE benchmarks against 11 physics-informed solvers, ReCAP achieves the lowest relative L squared error on every problem, reducing error by 10.7% -- 27.5% relative to the strongest competing result. These results suggest that physics-informed solvers need not merely learn their parameters---they can learn how their representational capacity should be organized.