Suppose you are working with a physical system that you mostly understand.

You know the governing equation. You trust it. Heat conduction, diffusion, elasticity, fluid flow — pick one. The math itself is not the issue. The equation has been written in textbooks for decades, implemented in solvers for years, and validated across experiments.

What you don’t know is a parameter sitting quietly inside that equation.

It could be thermal conductivity, Young’s modulus, a diffusion coefficient, viscosity, or a reaction rate. You know it exists, you know it matters, and you know the system behaves differently when it changes. But you cannot directly measure it.

What you can do is observe the system response. You can measure temperatures at a few points, displacements at some locations, velocities along a line. These measurements are limited, noisy, and incomplete — but they are all you have.

This is not an exotic problem. Engineers deal with this situation constantly.

Traditionally, this would be handled using inverse methods: parameter fitting, least-squares optimization, adjoint-based solvers, or carefully designed numerical loops tailored to a specific equation. These approaches work, but they are often tedious, problem-specific, and difficult to generalize.

Physics-Informed Neural Networks promise a different route. And inverse PINNs push that promise one step further.

What an Inverse PINN Actually Is

A standard, or forward, PINN has a clear and limited task. You already know the governing equation. You already know all the parameters inside it. The neural network’s job is simply to approximate a function that satisfies the physics, boundary conditions, and possibly some data.

An inverse PINN changes only one thing — but that change is fundamental.

Instead of assuming all parameters are known, you treat one or more of them as unknown and ask the network to learn them from data.

That’s it. No hidden machinery beyond that.

For example, instead of writing the heat equation as

\[u_t = k \, u_{xx}\]

with a known conductivity $k$, you say:

\[u_t = k \, u_{xx}, \quad \text{but } k \text{ is unknown}\]

The neural network now has two learning tasks at the same time. It must learn the solution $u(x,t)$, and it must also infer the parameter $k$ that makes the equation consistent with the observed data.

From a coding point of view, this looks almost trivial. You declare $k$ as a trainable variable and include it in the loss computation.

From an optimization point of view, this changes everything.

Why Inverse Problems Are Fundamentally Different

The difficulty with inverse PINNs does not come from neural networks. It comes from inverse problems themselves.

In a forward problem, the physics strongly restricts the solution. The governing equation, boundary conditions, and initial conditions leave very little freedom. Only a narrow class of functions can satisfy all constraints simultaneously.

In an inverse problem, that restriction weakens.

The same physics may allow many combinations of solution and parameters to explain the same limited observations. This is especially true when data is sparse, noisy, or poorly placed.

This phenomenon is known as non-uniqueness, and inverse PINNs do not eliminate it.

What makes things tricky is that the neural network can exploit this freedom in subtle ways. If the data is weakly sensitive to a parameter, the network may slightly distort the solution while adjusting the parameter to keep the PDE residual small. The data loss remains low. The physics loss remains low.

You can see this mathematically by thinking in terms of sensitivity. What the data actually “sees” is not the parameter $k$ itself, but how the solution changes when $k$ changes:

\[\frac{\partial u}{\partial k}\]

If this sensitivity is small in the regions where data is available, then very different values of $k$ can produce solutions that look nearly identical at the measurement points.

From the optimizer’s point of view, this creates flat valleys in the loss landscape. Moving the parameter slightly does not change the loss much, as long as the network adjusts the solution accordingly.

Everything looks correct — except the parameter.

This is why inverse PINNs often feel misleading rather than broken.

How Inverse PINNs Are Commonly Set Up

In practice, inverse PINNs are usually trained by minimizing a loss function that combines three components: the PDE residual, the boundary or initial condition error, and the mismatch between predictions and measured data.

Written more explicitly, this often looks like:

\[\mathcal{L} = \underbrace{\frac{1}{N_f}\sum_{i=1}^{N_f} \left| \mathcal{R}\big(u_\theta(x_i,t_i), k\big) \right|^2}_{\text{PDE residual}} + \underbrace{\frac{1}{N_b}\sum_{j=1}^{N_b} \left| u_\theta(x_j,t_j) - u_b \right|^2}_{\text{boundary / initial}} + \underbrace{\frac{1}{N_d}\sum_{m=1}^{N_d} \left| u_\theta(x_m,t_m) - u^{\text{data}}_m \right|^2}_{\text{data mismatch}}\]

Here, $\mathcal{R}(\cdot)$ denotes the differential operator defined by the governing equation, and $k$ appears explicitly inside it.

During training, gradients flow not only to the neural network weights $\theta$, but also to the physical parameter $k$:

\[\frac{\partial \mathcal{L}}{\partial k} \neq 0\]

From an implementation perspective, this is straightforward. Most autodiff frameworks handle this without complaint.

From a modeling perspective, it is dangerous if done without thought.

The optimizer does not “know” which variable represents physics and which represents approximation flexibility. It only sees gradients and loss reduction.

Why Loss Convergence Is Not Enough

One of the most common mistakes is trusting loss curves.

In inverse PINNs, a decreasing loss does not guarantee that the learned parameter is correct.

The reason is simple: the network has too much freedom.

If the neural network is flexible enough, it can produce a slightly wrong solution that still fits sparse data. At the same time, the parameter can shift to compensate so that the PDE residual remains small.

Mathematically, this means the loss can be minimized along directions where

\[\frac{\partial \mathcal{L}}{\partial \theta} \approx 0 \quad \text{and} \quad \frac{\partial \mathcal{L}}{\partial k} \approx 0\]

even though the true parameter value has not been recovered.

The optimizer has no reason to prefer the “true” parameter over a convenient compensating combination that also minimizes the loss.

This is not a numerical issue or a tuning problem. It is a structural property of the optimization landscape.

Identifiability: The Hidden Constraint

A key concept that often stays implicit in inverse PINN discussions is identifiability.

A parameter is identifiable only if changing that parameter produces observable changes in the data you have.

If those changes are weak or masked, no method — PINN or otherwise — can reliably recover the parameter.

Another way to see this is through the shape of the loss function with respect to the parameter. A well-identified parameter produces a sharp minimum: small changes in $k$ lead to noticeable increases in loss. A poorly identified parameter produces a flat or shallow minimum, where many values of $k$ look equally acceptable.

In practice, inverse PINNs often operate in this flat regime. The loss landscape with respect to the parameter resembles a plateau rather than a well-defined valley. Gradient-based optimizers can wander along this plateau for a long time without receiving a strong signal about where the “correct” value lies.

This shows up in very concrete ways. Trying to learn diffusivity from a nearly steady temperature field is extremely difficult. Trying to infer stiffness from displacement data with minimal load variation is unreliable. Trying to estimate viscosity from a flow with weak gradients often fails.

Inverse PINNs enforce physics, but they do not create information.

Strategies That Practitioners Actually Use

Because of these difficulties, practitioners rarely rely on naive end-to-end training.

One commonly used approach is pretraining. The network is first trained as a forward PINN using a guessed parameter value. This allows the solution shape to stabilize before the parameter is allowed to change. Only after that does parameter learning begin.

Another important practice is using separate learning rates. Physical parameters often require much smaller learning rates than neural network weights. Treating them identically can lead to oscillations, drift, or premature convergence to incorrect values.

Scaling and nondimensionalization matter more than many expect. Poor scaling can bias the optimizer toward adjusting the network instead of the parameter, or the other way around, without any obvious warning signs.

Finally, experienced users are cautious about learning many parameters at once. One unknown parameter is often manageable. Several usually are not, unless the data strongly constrains each of them.

None of these strategies guarantee success. But ignoring them almost guarantees failure.

A Simple Training Loop View

Stripped of frameworks and terminology, an inverse PINN training loop looks like this:

initialize network weights θ
initialize unknown physical parameter k

for each training step:
    predict solution u(x; θ)
    compute PDE residual using k
    compute error at measurement points
    total loss = physics loss + data loss

    backpropagate
    update θ
    update k

What matters here is not the loop itself, but the relative strength of gradients flowing into $\theta$ versus the network weights. In many cases, monitoring parameter gradients separately gives more insight than watching the total loss alone.

The challenge is deciding who is allowed to learn what, and when.

That control rarely emerges automatically.

Where Inverse PINNs Make Sense

Inverse PINNs make sense when the governing equations are well trusted, when classical inverse solvers are difficult to implement, and when geometry or multi-physics coupling makes adjoint methods cumbersome.

They are particularly attractive when data is sparse but informative, and when unifying solution learning and parameter estimation in a single framework reduces complexity.

But they are not replacements for judgment. Inverse PINNs struggle when parameters weakly influence the solution, when multiple parameter sets fit the data equally well, when noise dominates sparse measurements, or when the network is too flexible relative to the available data.

In such cases, inverse PINNs may produce estimates that look confident but are fundamentally unreliable.

Further exploration (optional)

For readers who want to see how inverse PINNs behave in practice — including parameter learning schedules, loss coupling effects, scaling choices, and how incorrect parameters can still produce low residuals — there is a hands-on walkthrough available as part of the PINNs Masterclass.

Learn More →