ml-debug/pinn/refs/heat_exchanger.md

Heat Exchanger PINN Notes

Project-specific notes for heat exchanger PINNs with phase change. Extracted from the general PINN skill to keep it domain-agnostic.

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Complexity ladder (heat exchanger example)

Level	Model	Assumptions	What it tests
0	Persistence: y(t) = y(t-1)	None	Does model capture any dynamics?
1	Exponential decay: T(t) = T_ss + (T_0 - T_ss) exp(-t/tau)	First-order response	Dominant thermal mode
2	Linear state-space / DMDc: fit A, B via OLS on finite differences (or Dynamic Mode Decomposition with control)	Linear dynamics	Near-linear regime
3	Pure data MLP (no physics loss)	NN can interpolate	If PINN doesn't beat this, physics constraint hurts
4	Effectiveness-NTU closed form	1D, constant Cp, steady state	Analytical sanity check
5	1D ODE with constant Cp (scipy solve_bvp)	1D, constant Cp	Classical numerical baseline
6	1D ODE with enthalpy (scipy solve_bvp)	1D, variable properties	Phase change handling
7	1D PINN with enthalpy	1D, learned U	Differentiable model, parameter estimation
8	2D PINN with radial maldistribution	2D, learned U + u_r	Full physics

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Hard BC example (heat exchanger)

For our heat exchanger: u_{r,tube}(z,r) = r(R-r) * NN(z,r) enforces zero radial velocity at r=0 and r=R by construction. Same principle as Sukumar & Srivastava 2022.

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Known failure modes

Degenerate U -> 0: When learning a heat transfer coefficient U jointly with the network, the optimizer may find U -> 0 (no heat transfer, flat profiles). This satisfies the PDE trivially.

• Fix: bound U in [100, 5000], initialize near a physically reasonable value (~500).
• Fix: penalize |dT/dz| being too small (the network should predict temperature change).

Sign errors in counterflow: The counterflow BVP in z-coordinates has both fluids' enthalpies decreasing in z:

• dh_t/dz = -UA'(T_t - T_s)/m_dot_t (tube cools, < 0 since T_t > T_s)
• dh_s/dz = -UA'(T_t - T_s)/m_dot_s (same sign! shell also decreases in z, but its flow is -z so it warms) The counterflow is encoded in BCs (h_t at z=0, h_s at z=L), not in opposing ODE signs. Common mistake: putting dh_s/dz = +q gives dh_s/dz > 0 (shell hot at z=0 exit, cold at z=L entry -- wrong).

NN vs scalar parameter: When a scalar (U) and a NN (temperature field) are optimized jointly, the NN is easier to update (more parameters, more gradient signal). The scalar gets stuck. Fix: use different learning rates, or estimate U separately first.

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Property Mappings (EoS for PINNs)

For heat exchangers with phase change, the T(h) mapping from REFPROP/GERG-2008 must be made differentiable.

Recipe:

1. Generate REFPROP data: T vs h at constant P, dense enough near the phase boundary (~200 points over T = [100, 300] K).
2. Fit a cubic spline to get ~20 minimal knots (reject redundant knots where the curve is nearly linear).
3. Build PCHIP on those knots for monotonicity guarantee (cubic spline can wiggle; PCHIP cannot).
4. Wrap for PyTorch autograd:

   class PchipFunction(torch.autograd.Function):
       @staticmethod
       def forward(ctx, h):
           T = pchip_interp(h.detach().numpy())  # scipy
           ctx.save_for_backward(h)
           return torch.tensor(T, dtype=h.dtype, device=h.device)
   
       @staticmethod
       def backward(ctx, grad_output):
           h, = ctx.saved_tensors
           dTdh = pchip_interp.derivative()(h.detach().cpu().numpy())  # scipy
           return grad_output * torch.tensor(dTdh, dtype=h.dtype, device=h.device)
   

5. Use float64 throughout. float32 loses precision near the phase boundary where dT/dh is small.

Why h -> T and not T -> h? The h(T) direction has a steep region near phase change where dh/dT = Cp -> infinity. The inverse T(h) is smooth and monotone everywhere. Always solve in h space, convert to T only for losses and visualization.

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Initial Conditions and Multi-Episode Training

IC is a loss term, but in practice ICs from real plant data are noisy and uncertain.

Options (in order of complexity):

1. Hard-code IC from first observed data point. Simplest; biased if sensor lags.
2. Learnable IC: optimize IC scalars jointly with the network. Adds ~n_states parameters. Works well with L-BFGS polish.
3. Soft IC loss: penalize ||u(t=0) - u_obs(0)||^2 as one loss term. Lets optimizer trade off IC vs physics fit.

Multi-episode training (e.g., ~10 cold restarts over a year):

• Each episode needs its own IC.
• Common pattern: learn per-episode IC vector ic_k while sharing the physics network across episodes.
• Warm-start trick: if you have a pure-data pre-fit (Level 3 baseline), initialize PINN data-loss weights from that fit.

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Domain decomposition (heat exchanger note)

Could split axially (cold end / hot end) if the solution has different character in different regions.

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Heat-exchanger-specific symptom table entries

Symptom	Likely cause
Loss decreases then explodes	Numerical instability in T(h) near phase boundary
Loss NaN	temperature outside EoS range
U converges to 0 or bound	Degenerate solution, check sign convention
PINN worse than pure-data MLP	Physics constraint fighting the data