After I published my earlier blog post on Ψ-NN (Physics Structure-Informed Neural Networks), I received a very kind comment from one of the authors. In that reply, they pointed me to two earlier projects from the same research line: AsPINN (Adaptive Symmetry-Recomposition Physics-Informed Neural Networks) and AtPINN (Adaptive Transfer Learning for PINN). At first glance, these names sound like “yet another two Physics-Informed Neural Networks (PINN) variants”. But after reading the papers, I think they are worth highlighting for a different reason: they reveal the motivation behind the whole sequence that ultimately led to Ψ-NN.
These are not academic details, but the difference between a PINN being a fragile research demo and becoming a robust modeling tool. Symmetry matters in PINNs because a lot of physics is not just “a PDE plus boundary conditions”. It is also structure: invariances, conservation laws, odd and even symmetry, sign symmetry, and geometric invariance. A vanilla PINN does not automatically learn these symmetries, even when the governing equations imply them. The network can approximate a symmetric solution, but it often does so inefficiently. It converges more slowly, it wastes parameters representing symmetric parts twice, and it can drift into slightly asymmetric solutions, which are physically wrong. This matters because PINNs are already difficult to train (I remember training PINN models for months while working for the German Aerospace Center (DLR); PINNs combined with Reinforcement Learning and CARLA were the reason I even bought my Desktop-PC at the time). Their optimization landscape is stiff, multi-objective, and sensitive to initialization. If the model is also allowed to explore function classes that violate symmetry, you are giving it more ways to fail. Symmetry is therefore one of the strongest priors you can inject into a physics-informed model.
This is the niche where AsPINN sits. The key idea is simple to state. Instead of hoping the PINN will learn symmetry implicitly, you build an architecture that can enforce symmetry explicitly through its structure. AsPINN uses symmetry blocks and an attention-style recomposition mechanism. Conceptually, this is the opposite of what most PINN practitioners do. The classic approach is to train a standard PINN, then add a symmetry penalty, or manually post-process the solution, or enforce symmetric sampling. AsPINN instead says that if symmetry is real, do not treat it as an optional regularizer. Make it part of the hypothesis space. The benefit is not only better final accuracy. The bigger gain is training stability and efficiency. When you remove symmetry-violating solutions from the search space, the optimizer stops wasting time. In engineering terms, AsPINN is not about learning more. It is about searching less.
The second bottleneck is transfer. In real scientific and engineering workflows, you rarely solve one PDE once. You solve families. You solve Burgers’ equation at different viscosities, heat diffusion at different conductivities, fluid flow at different Reynolds numbers, device models at different process corners, and the same geometry under different operating conditions. If you train a PINN from scratch every time (like I did at DLR before), you pay the full compute cost again and again. That makes PINNs hard to integrate into any serious design workflow. This is one reason PINNs often stay stuck in single-problem demos. The method might work, but it does not scale.
This is where AtPINN comes in. AtPINN focuses on a very practical question. If we already trained a physics-informed model on one PDE setting, how can we adapt it to a new setting quickly and reliably. The core contribution is an adaptive transfer procedure designed to avoid the common failure mode of naive fine-tuning, namely instability when the PDE parameters change. In practice, AtPINN tries to keep the optimization trajectory in a safe region of the loss landscape, rather than throwing the network into a new regime and hoping gradient descent will find its way. This is not just a trick to save time. It also affects correctness. Transfer methods can prevent the network from converging to a different spurious local minimum when parameters change. So while AsPINN is about reducing the hypothesis space using known structure, AtPINN is about reusing learned structure across regimes.
This is why, when I read the Ψ-NN first time I had the thought „I feel that this paper will be as successful if not even more than the PINN paper in the coming years“). AsPINN and AtPINN address two major pain points in physics-informed learning, but both still operate in a regime where the structure is either known, in the case of symmetry, or assumed to be reusable, in the case of transfer. Ψ-NN goes after the harder problem: what if we do not know what the structure is.
In my own case, this is also why Ψ-NN had such a direct impact. My problem was not “solve a PDE better”. My problem was whether we can take physics-informed learning and produce a deployable artifact for circuit simulation. That is what led to Ψ-HDL (Physics Structure-Informed Neural Networks for Hardware Description Language Generation), which I published recently in IEEE Access. The key extension was to take the Ψ-NN structure discovery pipeline and add a missing final step: deployment translation. In other words, Ψ-NN discovers structure in a network, and Ψ-HDL translates discovered structure into simulator-ready Verilog-A and SPICE-compatible behavioral compact models.
AsPINN and AtPINN are worth reading not because they are two more PINN variants, but because they make Ψ-NN look less like a one-off trick and more like the next step in a coherent research program + it helps understanding the broader research line around Ψ-NN.
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